Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

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���

62/87,21���

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����

�

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���

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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�

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

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� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 1

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

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�

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

�����

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

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� Check:

����

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�����

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 2

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

�����

�

62/87,21���

� Check:

�����

�

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� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

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� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 3

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

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� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 4

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

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� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

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����

62/87,21���

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����

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����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 5

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

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� Check:

����

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� Check:

����

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� Check:

�����

�

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� Check:

�����

�

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� Check:

����

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� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 6

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 7

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

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�����

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����

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����

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����

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� Check:

����

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 8

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

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���

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����

�

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� Check:

���

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� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

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� Check:

����

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� Check:

�����

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�

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�

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

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����

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����

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� Check:

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 9

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

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� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 10

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 11

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

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� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

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� Check:

����

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�

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�

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� Check:

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� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 12

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 13

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

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� Check:

����

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

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� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 14

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

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�

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

�����

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

�����

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�����

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 15

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 16

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

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���12 = ±7f ± 9

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

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����9x + 27 = ±72

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

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����45 = 7 ± 5n

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

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����0.6a + 9 = 14.4

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����3.6 ± 2.4m = 12

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����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 17

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

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�

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

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����

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�����

�

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����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 18

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 19

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

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� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

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����

62/87,21���

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����

62/87,21���

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����

62/87,21���

� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 20

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

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�����

�

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� Check:

����

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� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

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����

62/87,21���

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����

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����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 21

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

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� Check:

����

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� Check:

����

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� Check:

�����

�

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� Check:

�����

�

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� Check:

����

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� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 22

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 23

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

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� Check:

�����

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����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 24

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

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� Check:

���

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� Check:

����

�

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� Check:

���

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� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

�����

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�

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�

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� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 25

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 26

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

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� Check:

����

62/87,21���

� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 27

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

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� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

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� Check:

����

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� Check:

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�

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�

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� Check:

����

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� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 28

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 29

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

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� Check:

����

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�

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

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� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 30

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

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�

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

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� Check:

����

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� Check:

�����

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

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� Check:

�����

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�����

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 31

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 32

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

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���12 = ±7f ± 9

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

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����9x + 27 = ±72

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

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����45 = 7 ± 5n

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

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����0.6a + 9 = 14.4

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����3.6 ± 2.4m = 12

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����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 33

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

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�

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���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

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����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

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Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

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����

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�����

�

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����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 34

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 35

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

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� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

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����

62/87,21���

� Check:

����

62/87,21���

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����

62/87,21���

� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 36

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

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� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

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����

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����

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� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 37

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

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� Check:

����

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� Check:

����

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� Check:

�����

�

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� Check:

�����

�

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� Check:

����

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� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

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� Check:

����

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� Check:

����

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� Check:

����

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� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 38

2-3 Solving Multi-Step Equations

Solve each equation. Check your solution.���3m + 4 = ±11

62/87,21���

� Check:

���12 = ±7f ± 9

62/87,21���

� Check:

���

�

62/87,21���

� Check:

���

62/87,21���

� Check:

����

�

62/87,21���

� Check:

���

62/87,21���

� Check:

���NUMBER THEORY Twelve decreased by twice a number equals ±34. Write an equation for this situation and then find the number.

62/87,21���Let n = a number.

�

� The equation is 12 ± 2n = ±34, and the number is 23.

Twelve decreased by

twice a number

equals ±34.

12 ± 2n = ±34

���BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career?

62/87,21���Let h = the number of home runs Dave Winfield hit. � �

�

� Dave Winfield hit 465 home runs in his career.

175 fewer than twice the

number that Dave Winfield

has

equals the number of home runs

Hank Aaron has

2h ± 175 = 755

Write an equation and solve each problem.���Find three consecutive odd integers with a sum of 75.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75. �

� The integers are 23, 25, and 27.

����Find three consecutive integers with a sum of ±36.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is ±36. So, n + (n + 1) + (n + 2) = ±36. �

� The integers are ±13, ±12, and ±11.

Solve each equation. Check your solution.����3t + 7 = ±8

62/87,21���

� Check:

����8 = 16 + 8n

62/87,21���

� Check:

����±34 = 6m ± 4

62/87,21���

� Check:

����9x + 27 = ±72

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����

62/87,21���

� Check:

����FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget.

62/87,21���Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and the cost per minute is $0.15. So, 0.15m + 49.99 = 100. �

� Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.

Write an equation and solve each problem.����Fourteen less than three fourths of a number is negative eight. Find the number.

62/87,21���Let n = the number.

�

� The number is 8.

Fourteen less than

three fourths of a

number

is negative eight.

± 14 = ±8

����Seventeen is thirteen subtracted from six times a number. What is the number?

62/87,21���Let x = the number.

�

� The number is 5.

Seventeen is thirteen subtracted from six times a number.

17 = 6x ± 13

����Find three consecutive even integers with the sum of ±84.

62/87,21���Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive even integers is ±84. So, n + (n + 2) + (n + 4) = ±84. �

� The integers are ±30, ±28, and ±26.

����Find three consecutive odd integers with the sum of 141.

62/87,21���Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three integers. The sum of the three consecutive odd integers is 141. So, n + (n + 2) + (n + 4) = 141. �

� The integers are 45, 47, and 49.

����Find four consecutive integers with the sum of 54.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54. �

� The integers are 12, 13, 14, and 15.

����Find four consecutive integers with the sum of ±142.

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of the integers. The sum of the three consecutive integers is ±142. So, �Q + (n + 1) + (n + 2) + (n + 3) = ±142. �

� The integers are ±37, ±36, ±35, and ±34.

Solve each equation. Check your solution.����±6m ± 8 = 24

62/87,21���

� Check:

����45 = 7 ± 5n

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

62/87,21���

� Check:

�����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

����

62/87,21���

� Check:

Write an equation and solve each problem.����CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the

brothers?

62/87,21���Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96. �

� The brothers are 31, 32, and 33.

����VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?

62/87,21���To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y .�

�

The lava has flowed years or 7 years and 3 months.

Solve each equation. Check your solution.����±5x ± 4.8 = 6.7

62/87,21���

� Check:

����3.7q + 26.2 = 111.67

62/87,21���

� Check:

����0.6a + 9 = 14.4

62/87,21���

� Check:

����

62/87,21���

� Check:

�����

�

62/87,21���

� Check:

����3.6 ± 2.4m = 12

62/87,21���

� Check:

����If 7m ± 3 = 53, what is the value of 11m + 2?

62/87,21���To find the value of 11m + 2, first solve 7m ± 3 = 53 to find the value of m.�

� Now, replace m with 8 in the expression 11m + 2. �

� So, 11m + 2 = 90.

����If 13y + 25 = 64, what is the value of 4y ± 7?

62/87,21���To find the value of 4y ± 7, first solve 13y + 25 = 64 for y .�

� Now, replace y with 3 in the expression 4y ± 7. �

� So, 4y ± 7 = 5.

����If ±5c + 6 = ±69, what is the value of 6c ± 15?

62/87,21���To find the value of 6c ± 15, first solve ±5c + 6 = ±69 for c.�

� Now, replace c with 15 in the expression 6c ± 15. �

� So, 6c ± 15 = 75.

����AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park. c. Plot these points on a coordinate graph and describe what you see.

62/87,21���a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit, set the two expressions equal to each other and solve for x. �

� The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11visits. � b.

� c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table. Use a different colored point for the members and nonmembers.

Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11. Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.

Visits Cost for Members

Cost for Nonmembers

3 5(3) + 275 = 290

3(6 + 15 + 9) = 90

6 5(6) + 275 = 305

6(6 + 15 + 9) = 180

9 5(9) + 275 = 320

9(6 + 15 + 9) = 270

12 5(12) + 275 = 335

12(6 + 15 + 9) = 360

15 5(15) + 275 = 350

15(6 + 15 + 9) = 450

����SHOPPING At The Family Farm, you can pick your own fruits and vegetables.

a. The cost of a bag of potatoes is $1.50 less than of the price of apples. Write and solve an equation to find the

cost of potatoes. b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.

62/87,21���a. Let a = the cost of a bag of apples and p � �WKH� cost of a bag of potatoes. �

�

� The cost of a bag of potatoes is about $2.00. � b. Let z = the price of zucchini and w = the price of winter squash.

�

� The cost of zucchini is $1.97. � c. Let p = the cost of a pumpkin and b = the cost of blueberries. �

� An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b ± 0.98.

The cost of a bag

of potatoes

is $1.50 less than

of the price

of apples. p =

The price of each zucchini

is 3 times the

price of winter squash

minus $7.

z = 3w ± 7

The cost of a

pumpkin

is 2 times the cost of

blueberries

minus 0.98

p = 2 � b ± 0.98

����OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and explain the solution in the context of the problem.

62/87,21���Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T±shirt. How much is the T±shirt? �

� The T±shirt costs $10.

����CHALLENGE Solve each equation for x. Assume that a������ D��� � E���

� F���

62/87,21���� D����

� � E����

� F����

����Determine whether each equation has a solution. Justify your answer. � a.

� b.

� c.

62/87,21���a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have a solution. � b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 ± b. If we subtract 1 from each side, we are left with b = ±b which is true only when b = 0. Therefore, the equation has a solution, 0. � c. For any fraction to equal 1, the numerator and denominator must be equal. So, c ± 5 must equal 5 ± c. If we add c+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction becomes or which is undefined. Therefore, the original equation does not have a solution.

����CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer.

62/87,21���The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two odd numbers will always sum to an even number, and the sum of this even number and the third odd number will DOZD\V�EH�RGG�� � Test a few examples: � 3 + 5 + 7 = 15 9 + 13 + 17 = 39 11 + 19 + 33 = 63 � The algebraic proof of this statement is beyond the scope of this course.

����WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-stepequation.

62/87,21���Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example, in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by 4.

����Which is the best estimate for the number of minutes on the calling card advertised below?

A 10 min B 20 min C 50 min D 200 min

62/87,21���To estimate the number of minutes on the calling card, divide $10 by $0.05. ����·������� ���� So, there are about 200 minutes on the calling card. Choice D is the correct answer.

����GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56cm. What is the perimeter of the larger triangle in centimeters?

62/87,21���Use a proportion to find the perimeter of the larger triangle.�

� The perimeter of the larger triangle is 84 centimeters.

����Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 ft3 = 7.5 gal) F 37.7 min G 7 h H 25.4 h J 35.3 h

62/87,21���To find about how long it will take to drain the pool, first calculate the amount of water in the pool. �

� There are about 1413ft3 of water in the pool. Because 1 ft3 = 7.5 gallon, then �

. Use the equation t = w�·�r, where t = time to drain the pool, w�� �DPRXQW�RI�ZDWHU�LQ�WKH�SRRO�DQG�r = rate water is pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take ��������JDOORQV�·���JDOORQV�PLQXWH�RU�DERXW��������PLQXWHV�WR�GUDLQ�WKH�SRRO���7R�FKDQJH�WKLV�WR�KRXUV��GLYLGH��������minutes by 60 minutes which is 35.325 �����K���&KRLFH�)�LV�WKH�FRUUHFW�DQVZHU� � �

����STATISTICS Look at the golf scores for the five players in the table.

Which of these is the range of the golf scores? A 10 B 25 C 35 D 40

62/87,21���To find the range subtract the least score from the greatest score.103 ± 78 = 25 � Choice B is the correct answer.

����GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas?

62/87,21���Let x be the number of miles a luxury car travel on a gallon of gas. �

� A luxury car travel 24 miles on a gallon of gas.

Miles for a 4-cylinder/ one

gallon

is 10 miles more than

Miles for an 8-cylinder/one

gallon 34 = 10 + X

�

����DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male deer born. How many male deer were born that year?

62/87,21���Let m = the number of male deer that were born. �

� 1379 male deer were born that year.

The number of female deer

is 93 fewer than the number of male deer

born. 1286 = m ± 93

�

Translate each equation into a verbal sentence.����f ± 15 = 6

62/87,21���f ± 15 = 6

A number f minus 15 is 6.

����3h + 7 = 20

62/87,21���3h + 7 = 20

Three times a

number h

is increased

by

7 to equal 20.

����k2 + 18 = 54 ± m

62/87,21���k2 + 18 = 54 ± m A

number k is

squared

and added

to

18 to equal 54 decreased by

m.

����3p = 8p ± r

62/87,21���3p = 8p ± r

Three multiplied by a number p

is the same as

the difference of 8

times p and r.

���� t + = t

62/87,21���

t

+

= t

Three fifths of t

added to is t.

���� v = v + 4

62/87,21���

v

= v

+ 4

The product of

�DQG�v

is equal to the product of

and v

plus 4.

����GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation.

62/87,21���46% written as a decimal is 0.46. �

� �7KHQ��P = 0.46E.

Surface Area of thePacific Ocean = percent ā Surface Area of

the EarthP = 0.46 � E

Find the value of n in each equation. Then name the property that is used.����1.5 + n = 1.5

62/87,21���Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.

����8n = 1

62/87,21���

Because 8 = 1, n = . This is the Multiplicative Inverse.

����4 ± n = 0

62/87,21���Because 4 ± 4 = 0, n = 4. This is the Additive Inverse.

����1 = 2n

62/87,21���

Because 1 = 2 , n = . This is the Multiplicative Inverse.

Evaluate each expression.����5 + 3(42)

62/87,21���

����

62/87,21���

����[5(1 + 1) ]3

62/87,21���

����[8(2) ± 42 ] + 7(4)

62/87,21���

eSolutions Manual - Powered by Cognero Page 39

2-3 Solving Multi-Step Equations