math 093 hybrid workbooks spring 2010
TRANSCRIPT
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Chapter 1 Equations, Inequalities and Applications
1.1 The Addition Property of Equality Learning Objectives:
1. Use the Addition Property of Equality to solve linear equations. 2. Simplify an equation and then use the Addition Property of Equality. 3. Write word phrases as algebraic expressions. 4. Key Vocabulary: solving, equivalent equations, addition property of equality.
A. Using the Addition Property Definitions:
1. Linear Equation in One Variable • is an equation of the form CByAx =+ , where BA, and C are any real numbers
and 0≠A . 2. Addition Property of Equality:
• If ba = , then cbca +=+ , where ba, and c are any real numbers. 3. Distributive Property:
• ( ) acabcba +=+ and ( ) acabcba −−=+− , where ba, and c are any real numbers.
Example 1. Solve each equation. Check each solution. 1. 518 −=− t
2. 43
32
−=+a
B. Simplifying Equations Steps to Simplify Equations:
1. Simplify each sides of equation as much as possible. 2. If an equation contains parentheses, use the distributive property to remove the parentheses. 3. Using the proper of equality to solve the resulted equation.
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Example 2. Solve each equation. 1. 103658 +−=++− xxx 2. 5.171.527.249.14 ++=+−+− aaa
3. 1 1 5 16 3 6 2
x x− − = +
4. ( ) ( ) 315523 −=+−−− yy
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C. Writing Algebraic Expressions Algebraic Expressions—are expressions that contain variable. Example 3. Write each algebraic expression described. 1. Two numbers have a sum of 72. If one number is x, expresses the other number in terms of x.
2. A 6‐foot board is cut into two pieces. If one piece is y feet long, express the other length in terms of y.
3. On a recent car trip, Raymond drove x miles on day one. On day two, he drove 170 miles
more than he did on day one. How many miles, in terms of x, did Raymond drive for both days combined?
1.1 Exercise Solve each equation.
1. 158 =+x 2. 129 =−y 3. 7.52.4 −=−r
4. 41
85
=+ k 5. xx 151114 =− 6. 23359 −=− xxx
7. 211812 +=− yy 8. 9201721 −=− pp 9. 1439511 +=−+ yyy
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10. xxxx 2716351224 ++=+−+ 11. 1.3118.137.27 +=−+− xxx
12. 65
127
32
125
−−=− yy 13. ( ) kk 71461219 +=−−
14. ( ) 12354 +=− xx 15. ( ) xx 293 −=−−
16. ( ) ( ) zzz 23827 =−+−
17. Two angles have a sum of o146 . If one angle is ox , express the other angle in terms of ox .
18. A 12‐foot board is cut into two pieces. If one piece is y feet long, express the other length in terms of y. 19. From Chicago, it is 31 more miles to Montreal than it is to New York City. If it is m miles to New York, express the distance to Montreal in terms of m.
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1.2 The Multiplication Property of Equality Learning Objectives:
1. Use the multiplication property of equality to solve linear equations. 2. Use both the addition and multiplication properties of equality to solve linear
equations. 3. Write word phrases as algebraic expressions. 4. Key Vocabulary: reciprocal, consecutive integers.
A. Using the Multiplication Property Multiplication Property of Equality:
• If ba = then bcac = where ba, and c are any real numbers. Example 1. Solve the following linear equations. 1. 248 −=− x
2. 5.211
=−
y
3. xx 5.15.329.18.5 −−=+− 4. ( ) ( )248144 −−−=−x
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B. Writing Algebraic Expressions Example 2. Write each algebraic expression described. Simplify if possible. 1. If x represents the first of two consecutive even integers, express the sum of the two integers
in terms of x. 2. If y represents the first of three consecutive odd integers, express the sum of the first and
third integer in terms of y.
1.2 Exercise Solve each equation.
1. 488 =− x 2. 37−=− x 3. 18119
−=y
4. 212
=d 5. 0
17=
−p 6. 75.103.4 =x
7. 14157
=− y 8. 3156 =−x 9. 819 =+− x
10. 41712 −=+− y 11. 197.019 =− k 12. 37
324 =+− x
13. 4125
=+b 14. 032487 =+−− zz 15. xx 5142615 −=+
16. 102.05.04 −=− xx 17. zzzz 6758 −−=− 18. If x is the first of three consecutive even integers, write their sum as an algebraic expression in x. 19. Houses on one side of a street are all numbered using consecutive odd integers. If the first house on the street is numbered x, write an expression in term of x for the sum of five house numbers in a row.
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1.3 Further Solving Linear Equations Learning Objectives:
1. Apply the general strategy for solving a linear equation. 2. Solve equations containing fractions and decimals 3. Recognize identities and equations with no solution. 4. Key Vocabulary: least common denominator (LCD), identity, no solution.
Steps for Solving Linear Equations
1. If an equation contains fractions, multiply both sides by the LCD to clear fractions. 2. If an equation contains decimal, multiply both sides by the power of ten according to the numbers of the decimal digit. 3. If and equation contains parentheses, use distributive property to remove the parentheses. 4. Simplify each side of the equation by combining like terms. 5. Get all variable terms on one side and all numbers on the other side by using the addition or the multiplication property of equality. 6. Check the solution by substituting the result into the original equation.
Example 1. Solve the following linear equations. 1. ( ) 4156 =−− aa 2. ( ) ( )45323 +=−− xx
3. 4
514
56 yy−=+
+−
9
4. ( ) 570150006.005.0 =−+ xx
5. 16
233
+=−xx
1.3 Exercise Solve each equation. 1. xx 158712 +=− 2. ( ) xx 10743 −=−− 3. ( ) ( ) 837135 ++=− nn 4. ( ) ( ) 1435526 −=−−+ xx 5. ( ) ( ) 283247 =+− xx 6. ( ) 171154 =−−− x
7. 63148 −=+ xx 8. ( ) aaa 31119 +=−+− 9. 92
32
94
=−x
10. yy =−1083 11. 6
54
3−=−
xx 12. ( ) ( )4
526
34 −=
− kk
13. ( ) ( )16832.08052.076.0 =+x 14. ( ) ( )20007.012.002.020044.0 +=+− yyy
15. ( ) 427
210−=
+ yy 16. 21
32
65
=−x 17. ( ) ( )xx 10541058 +=−
18. ( ) ( ) 3466349 −−=− yy 19. The perimeter of a geometric figure is the sum of the lengths of its sides. If the perimeter
of a trapezoid is 29 cm, and the length of the sides are ( )52,,2 +xxx and x cm, find the length of each side.
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1.4 An Introduction to Problem Solving Learning Objectives:
1. Translate a problem to an equation, and then use the equation to solve the problem. 2. Key Vocabulary: understand, translate, solve, and interpret.
General Strategy for Problem Solving
1. Understand the problem by doing the following • Read and reread the problem carefully. • Choose a variable to represent the unknown quantities. • Construct a drawing if needed.
2. Translate the problem into an equation. 3. Solve the equation using algebra. 4. Verify the solution (Check if the answer making sense).
Example 1. Solve each word problem. 1. Eight is added to a number and the sum is doubled, the result is –11 less than the number.
Find the number. 2. The difference between two positive integers is 42. One integer is three times as great as the
other. Find the integers. 3. When you open a book, the left and right page numbers are two consecutive natural
numbers. The sum of their page numbers is 349. What is the number of the page that comes first?
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4. A college graduating class is made up of 450 students. There are 206 more girls than boys. How many boys and girls are in the class?
5. A 22‐ft pipe is cut into two pieces. The shorter piece is 7 feet shorter than the longer piece.
What is the length of the longer piece? 6. A triangle has three angles, A, B, and C. Angle C is 18° greater than angle B. Angle A is 4
times angle B. What is the measure of each angle? (Hint: The sum of the angles of a triangle is 180°).
1.4 Exercise Solve each word problem.
1. The sum of five times a number and 87 is equal to the difference between six times a number
and41 . Find the number.
2. Eight times the sum of a number and 2− is the same as nine times the number. Find the number.
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3. Twice the difference of a number and seven is equal to five times the number plus one. Find the number.
4. If the sum of a number and −3 is doubled the result is 12 times the number. Find the number. 7. Sue makes twice as much money as Tom. If the total of their salaries is $78,000, find the salary of each. 8. Peggy Fleming won two more U.S. Figure Skating Championships than Dorothy Hamill. If the total championship for both is 8, find how many each won. 9. A 30‐inch board is to be cut into three pieces so the second piece is twice as long as the first piece and the third piece is three times as long as the first piece. Find the length of all three pieces. 10. A 56‐inch board is to be cut into three pieces so the second piece is three times as long as the first piece and the third piece is four times as long as the first piece. Find the length of all three pieces. 11. A carpenter gave an estimate of $980 to build a cover over a patio. His hourly rate is $28 and he expects to need $560 in materials. How many hours does he expect the job to take? 12. A mechanic charged $239 to repair a car, including $107 in parts and 6 hours of labor. How
much does she charge per hour for labor?
13. An appliance repairman charges $75 to come to your house and $35 per hour. During one week, he visited 9 homes and his total weekly income was $1305. How many hours did he spend working on appliances?
14. Two angles are supplementary if their sum is 180°. One angle measures four times the measure of an angle supplementary to it. Find the measure of the angle. 15. Two angles are complementary. The second angle is six less than three times the first. Find the two angles. 16. The height of a soup can is 3.5 cm more than its diameter. If the sum of the height and the diameter is 16.5, find each dimension. 17. Find two consecutive even integers so that three times the smaller is 40 more than two times the larger. 18. Find three consecutive odd integers whose sum is negative 93. 19. Karl’s license plate is four consecutive integers with a sum of 26. What is his license plate number? 20. The sum of the angles of any four‐sided polygon is 360°. If the measures of the angles of a four‐sided polygon are four consecutive odd integers, find the measure of each angle.
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1.5 Formulas and Problem Solving Learning Objectives:
1. Given a formula and values, solve for the unknown. 2. Solve a formula or equation for one of its variables. 3. Solve word problems. 4. Key Vocabulary: formula, perimeter, area, volume.
A. Using Formulas to Solve Problems Formula—describes a known relationship among quantities. Example 1. Substitute the given values into each given formula and solve for the unknown
variable.
1. Distance Formula: rtd = ; t = 9, d = 63
2. Volume of a pyramid: 1 ;3
V Bh= V= 40, h = 8
B. Solving a Formula for a Variable Steps for Solving Equations for a Specified Variable
1. Multiply both sides of equation to clear fractions if they occur. 2. Use the distributive proper to remove parentheses if they occur. 3. Simplify each side of the equation by combining like terms if needed. 4. Get all terms containing the specified variable on one side and all other terms on the
other side by using the addition property of equality. 5. Get the specified variable alone by using the multiplication property of equality.
Example 2. Solve each formula for the specified variable.
1. 12
A bh= for b.
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2. ( )radL ++= π2 for a. Example 3. Solve
1. Convert the record high temperature of 102°F to Celsius. (Use the formula F = 9 325
C + )
2. You have decided to fence an area of your backyard for your dog. The length of the area is 1 meter less than twice the width. If the perimeter of the area is 70 meters, find the length and width of the rectangular area.
1.5 Exercise Substitute the given values into the formula and solve for the unknown variable. 1. rtD = when 272=D and 68=r
2. hrV 2
31π= when 1.47=V and 3=r (Leave the answer in term of π.)
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3. 3
34 rV π= when 5=r (Leave the answer in term of π.)
4. ( )hbBA +=21 when 8,5.45 == BA and 7=h .
5. ( )hbBA +=21 when 15,177 == BA and 11=b .
6. 12
12
xxyym
−−
= when 3,5,15 212 ==−= xyy and 81 =x
7. 12
12
xxyym
−−
= when 7,3,31
21 === xym and 21 −=x
Solve the following applications. 8. Wade has 126 inches of 1‐inch wide bias tape for a border on a rectangular banner. If the banner needs to be 48 inches long, what is the maximum width it could be? 9. It is 328 miles from Guymon to Tulsa. How long should it take Manuella to drive from Guymon to Tulsa if she averages driving 50 miles per hour? Use the formula rtd = .
10. The formula 3259
+= CF can be used to convert temperatures in degrees Celsius to degree
Fahrenheit. Convert Istanbul, Turkey’s 28° C average daily high in July to Fahrenheit. 11. Find how many piranhas you can put in a cylindrical tank whose diameter is 4 feet and whose height is 1.25 feet if each piranha needs 1.5 cubic feet of water. 12. Which has more pizza, one 20‐inch pizza or two 15‐inch pizzas, if the size indicates the diameter of a round pizza? Solve each formula for the specified variable.
13. dcbaP +++= for d 14. 1074 =− yx for y 15. AhV31
= for A
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1.6 Solving Linear Inequalities Learning Objectives:
1. Graph inequalities on a number line. 2. Use the addition property of inequality to solve inequalities. 3. Use the multiplication property of inequality to solve inequalities. 4. Use both properties to solve inequalities. 5. Solve problems modeled by inequalities. 6. Key Vocabulary: inequality, <, <, >, >, addition property of inequality, multiplication
property of inequality, at least, no less than, at most, no more than, is less than, is greater than.
A. Graphing Inequalities on a Number Line Inequality—is a statement that contains <, <, >, > symbols. Example1. Graph each inequality on a number line. 1. 5−≤x
2. m≤−23
3. 05 ≤<− t B. Solving the Inequalities using the Addition and Multiplication Property of Inequality Properties of Inequalities—Let a, b and c be real numbers, then 1. Addition Property:
• If ba < , then cbca +<+ and If ba > , then cbca +>+ . 2. Positive Multiplication Property: (c is positive)
• If ba < , then bcac < and If ba > , then bcac > . 3. Negative Multiplication Property: (c is negative)
• If ba < , then bcac > and If ba > , then bcac < .
0 2 4 -2 -4 6 -6
0 2 4 -2 -4 6 -6
0 2 4 -2 -4 6 -6
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CAUTION! If multiply or divide by a negative number, the inequality sign change to opposite. Example 2. Solve each inequality. Graph the solution set. 1. 1524 +−>−− aa
2. 953
≤− x
3. ( ) 2421218 +−≥−− yy
4. ( ) ( )3712
218
+>+ xx
0 2 4 -2 -4 6 -6
0 10 20 -10 -20 30 -30
0 2 4 -2 -4 6 -6
0 2 4 -2 -4 6 -6
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C. Solving Applications Involving Inequalities Key words: Is less than means < At most means ≤ Is greater than means > At least means ≥ No more than means ≤ Not equal to means ≠ Is less than or equal to means ≤ Is greater than or equal to means ≥ Example 2. Solve the following. 1. Eight more than twice a number is less than negative twelve. Find all numbers that make
this statement true.
2. One side of a triangle is six times as long as another sides, and the third side is 8 inches long. If the perimeter can be no more than 106 inches, find the maximum lengths of the other two sides.
1.6 Exercise Graph each on a number line.
1. 2−>x 2. 23
≤y 3. 13 <≤− x
Solve each inequality. 4. 46 −≥−x 5. 6527 +<− xx 6. 4534 −<− xx
7. 155 ≥− x 8. 1.27.0 −>− y 9. xxx 257 −≥−
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10. ( )52354 +<+− xx 11. ( ) ( )353672 −<− xx
12. ( ) ( ) 221245328 +−≥−+ xxx 13. ( ) ( ) 3124311537 +−≤−− xxx
14. ( ) ( ) 125483 +−−≥+−− xxx 15. ( ) ( ) 135517 −−>−−− xxx
16. ( ) ( )12413
61
−<+− xxx
Solve the following 17. Nine more than four times a number is greater than negative fourteen. Find all numbers that make this statement true. 18. Miranda needs an average of at least 90 to get an A in a course. She has earned scores of 82,
87 and 94 on her tests. The final exam counts as two tests. What score does she need on the final to get an A?
19. Tamara scored an 86 and a 92 on her last two math exams. What must she score on her third
exam to have an average of at least a 93?
20. Alex has at most 90 yards of fencing available to enclose a rectangular garden. If the width of the garden is to be 15 yards, find the maximum length that the garden can be.
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Chapter 2 Graphs and Linear Functions 2.1 Linear Equations and Their Solutions Learning Objectives: 1. Plot ordered pairs of numbers on the rectangular coordinate system. 2. Graph paired data to create a scatter diagram. 3. Find the missing coordinate of an ordered pair solution, given one coordinate of the pair. 4. Key Vocabulary: ordered pair, origin, quadrant, xaxis, yaxis, rectangular coordinate system,
coordinate plane, xcoordinate, ycoordinate, paired data, scatter diagram, solution of an equation in two variables.
Linear Equation—is an equation of the form cbyax =+ or bmxy += , where a, b, and c are any
real numbers. m is the slope and ( )b,0 is the y‐intercept. Solutions of Equations—is an ordered pair ( )yx, that satisfies the given equation meaning
when substitute the given ordered pair into the given equation will result a true statement.
Example 1. Complete each ordered pair so that it is a solution of the given linear equation. 1. 62 =+ yx ; ( ),2
2. 231
−= xy ; ⎟⎠⎞
⎜⎝⎛ −
31,
Example 2. Complete the table for the equation 131
−= xy .
x y
3−
0
0
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2.1 Exercise Complete each ordered pair so that it is a solution of the given linear equation.
1. 27 =− yx ; ( ),2 2. ⎟⎠⎞
⎜⎝⎛=+ ,
61;11312 yx
Complete the table of values for each given linear equation. 3. 824 =− yx 4. 2−=x
x y
0 0 2
x y
0 4− 3
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2.2 Graphing Linear Equations by Plotting Points Learning Objectives:
1. Graph a linear equation by finding and plotting ordered pair solutions. 2. Graph a linear equation and use the equation to make predictions. 3. Key Vocabulary: linear equation in two variables, graph of the equation, horizontal line,
vertical line.
Example 1. For each equation, find three ordered pair solutions. Then use the ordered pairs to graph the equation. 1. 842 =+− yx
2. 231
−−= xy
4. 4−=x
y
x
5
55−
5−
y
x
5
55−
5−
y
x
5
55−
5−
23
5. 3=y
2.2 Exercise For each equation, find three ordered pair solutions by completing the table. Then, use the ordered pairs to graph the equation. 1. 7=− yx 2. xy 3−= 3. 82 +−= xy Graph each linear equation. Label at least three points on the graph grid. 4. 1535 += xy 5. yx 6= 6. 1052 =− yx
7. xy41
−= 8. 62 =+− yx 9. 432
−= xy
Write the statement as an equation in two variables. Then graph the equation. 10. The y‐value is 6 less than the x‐value. 11. The sum of x and y is 7.
x y
0 3 0
x y
0 2 3
x y
0 1 5
y
x
5
55−
5−
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2.3 Graphing Lines Using Intercepts Learning Objectives:
1. Identify intercepts of a graph. 2. Graph a linear equation by finding and plotting intercept points. 3. Identify and graph vertical and horizontal lines. 4. Key Vocabulary: xintercept, yintercept, vertical line, horizontal line.
A. Graphing Lines Using Intercepts 1. The xintercept of a line is the point where the graph crossing the x‐axis.
To find the xintercept • Let y = 0, then solve for x. Ordered pair for x‐intercept: ( )0,a
2. The yintercept of a line is the point where the graph crossing the y‐axis. To find the yintercept
• Let x = 0, then solve for y. Ordered pair for y‐intercept: ( )b,0 Steps to Graph a Line Using the Intercepts. 1. Find the x‐intercept ( )0,a . 2. Find the y‐intercept ( )b,0 . 3. Graph the points ( )0,a and ( )b,0 , then connect them with a line. Example 1. Graph and label at least two points on the graph grid. 1. 1025 =+ yx 2. 03 =+ yx
y
x
5
55−
5−
y
x
5
55−
5−
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B. Graphing Vertical and Horizontal Lines 1. The Graph of ax = is a vertical line with x‐intercept ( )0,a .
2. The Graph of by = is a horizontal line with y‐intercept ( )b,0 . Example 3. Graph and label at least two points on the graph grid. 1. 033 =−x
2. 024 =+ y
2.3 Exercise Identify the intercepts and intercepts points. 1.
y
x
5
55−
5−
y
x
5
55−
5−
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2.
3.
4. Graph the line with x‐intercept at −8 and y‐intercept at 6. Graph each linear equation by finding x‐ and y‐intercepts. Label the x‐and y‐ intercepts on the graph grid. 5. 5=− yx 6. 63 −= yx 7. 1863 =− yx 8. xy −= 9. 5=y 10. 6−=x 11. Two lines in the same plane that do not intercept are called parallel lines. Graph the line x = − 2. Then graph a line parallel to the line x = − 2 that intersects the x‐axis at 3. What is the
equation of this line?
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2.4 The Slope of a Line Learning Objectives:
1. Find the slope of a line given two points of the line. 2. Find the slope of a line given its equation including horizontal and vertical lines. 3. Compare the slopes of parallel and perpendicular lines. 4. Slope as a rate of change. 5. Key Vocabulary: slope, rise, run, zero slope, undefined slope.
A. The Slope of Two Points
The slope m if the line going through the points ( )11 ,yx and ( )22 ,yx where 21 xx ≠ is given by B. The Slopes of Vertical and Horizontal Lines. 1. Vertical line ax = has _______________________________________________________________. 2. Horizontal line by = has _______________________________________. Example 1. Find the slope of the line going through 1. ( ) ( ).3,2and4,3 −− 2. ( ) ( ).1,3and1,4 − 3. ( ) ( ).1,2and4,2 −−
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C. The Slopes of Equations bmxy += Slope of bmxy += is ______________________. The y‐intercept is ___________________. Steps of Finding a Slope from the Equation: 1. Write the given equation in the form _____________________________________. 2. Identify the slope and y‐intercept. Example 3. Find the slopes and the y‐intercept of the following lines. 1. 423 =− yx 2. 632 =+ yx D. Finding Parallel and Perpendicular Lines
1. Parallel Lines • Two lines are parallel if 21 mm = but 21 bb ≠ .
2. Perpendicular Lines
• Two lines are Perpendicular if ( )( ) 121 −=mm or 1
21
mm −
= or 2
11
mm −
=
Example 4. Decide whether the pair of lines is parallel, perpendicular or neither.
1. 842
32=−
=−yx
yx
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2. 5363
=−=+
yxyx
2.4 Exercise
Find the slope of each equation. 1.
2.
3.
Find the slope of the line that goes through the given points. 4. ( )1,2 and ( )5,4 5. ( )3,7− and ( )6,7 −− 6. ( )2,8 − and ( )3,5 −
7. ( )9,6 −− and (−7, −10)
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Find the slope of each line.
8. 512 +−= xy 9. 1174 =− yx 10. 05 =+y 11. yx 8−=
Determine whether the lines are parallel, perpendicular, or neither.
12. 14242
=−=+
yxyx
13. 1234843
−==−
xyyx
14. 9575
=+=+
yxyx
Use the points given, (a) find the slope of the line parallel and (b) find the slope of the line perpendicular to the line through each pair of points. 15. ( )3,8 and ( )6,14 16. ( )6,5 −− and ( )9,7
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2.5 Equations of Lines Learning Objectives
1. Use the slope‐intercept form to write an equation of a line. 2. Use the slope‐intercept form to graph a linear equation. 3. Use the point‐slope form to find an equation of a line given its slope and a point on the
line. 4. Use the point‐slope form to find an equation of a line given two points on the line. 5. Use the point‐slope form to solve word problems. 6. Key Vocabulary: standard form, slopeintercept form, pointslope form.
A. Equations of Lines
1. Standard Form: CByAx =+ 2. Slopeintercept Form: bmxy += ; m = slope; ( )b,0 = y‐intercept 3. PointSlope Form: )( 11 xxmyy −=− ; ( )11, yx = given point; m = slope 4. Horizontal Line: by = ; 0=m , ( )b,0 = y‐intercept; x‐intercept = none 5. Vertical Line: ax = ; m = undefined; ( )0,a = x‐intercept; y‐intercept = none
Example 1. Find the slope‐intercept equation of the line that goes through the point ( )4,2 − and has a slope 3−=m , then graph.
Example 2. Find the slope‐intercept equation of the line having slope 5 and y‐intercept 4− ,
then graph.
y
x
5
55−
5−
y
x
5
55−
5−
32
B. Finding the equation of lines given two points Steps for finding the slopeintercept equation of a line given two points 1. Find the slope. 2. Find the equation of the line by first using the point‐slope form, and then write the equation
in the form of .bmxy += . 3. To graph, plot the given points ( )11 ,yx , ( )22 ,yx and joint them with a line. Example 3. Find the slope‐intercept equation of the line going through the points ( ) ( )4,3and2,5 −− , then graph.
2.5 Excise Write the equation of each line in the form bmxy +=
1. 4,32
−== bm 2. 4,81
=−= bm 3. 0,127
== bm 4. 9,0 −== bm
Use the slope‐intercept form to graph each equation. Label at least two points on the graph grid.
5. 32 +−= xy 6. 253
−= xy 7. 43 =+ yx 8. 823 =− yx
Find the slope‐intercept equation of the line with given slope and passing through the given point.
9. ,9=m through ( )1,3 10. ,7−=m through ( )6,1 −− 11. ,81
−=m through ( )5,16−
Find the equation of the line passing through each pair of points. Write the equation in the form CByAx =+
12. ( )7,4 and ( )12,5 13. ( )7,3 − and ( )11,3 −− 14. ( )2,9− and ( )0,0
y
x
5
55−
5−
33
A certain type of notebook earned a stationary company $12,000 in profit the first year and $22,000 the third year. 15. Assume the relationship between years on the market and profit is linear. Use ordered pairs of t, years on the market, and p, profit to write an equation of the relationship. 16. Use the equation to predict the profit the fifth year.
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2.6 Introduction to Functions Learning Objectives:
1. Identify relations, domains, and ranges. 2. Identify functions. 3. Use the vertical line test. 4. Use function notation. 5. Key Vocabulary: relation, domain, range, function, vertical line test, function notation.
A. Identifying Relations, Domains and Ranges: Definition: Relation—is a set of ordered pairs. Domain of the relation—is the set of all possible x‐values. Range of the relation—is the set of all possible y‐values. Types of Functions 1. Linear Function: bmxy += ; Domain: all real numbers; Range: all real numbers 2. Quadratic Function: cbxaxy ++= 2 ; Domain: all real numbers
3. Rational Function: QPy = ; Domain: all real numbers except 0=Q
Range: all real numbers except 0=y Example 1. Find the domain and range: 1. ( ) ( ) ( ){ }3,1,6,4,4,6 −−−=T
2. ( )⎭⎬⎫
⎩⎨⎧
+=
11,
xyyx
3. ( ){ }2, xyyx =
35
B. Identifying Functions Function—is a set of ordered pairs in which each domain value has exactly one range value; that
is, no two different ordered pairs have the same first coordinate. Function Notation: ( )xf read “ f of x” or “f evaluate at x” Example 2. Determine whether the relations are functions: 1. ( ) ( ) ( ){ }3,1,6,4,4,6 −−−=T 2. ( ) ( ) ( ) ( ){ }3,4,3,1,6,4,4,6 −−−=T 3. ( ){ }13, −= xyyx 4. ( ){ }2, xyyx = 5. ( ){ }2, yxyx = C. Using the Vertical Line Test Vertical Line Test—if a vertical line can be drawn so that it intersects a graph more than once,
then graph is not the graph of a function. Example 3. Determine whether the graph is that of a function. 1. 2.
y
x
5
–
–5
y
x
5
–
5 –5
36
D. Evaluating Functions Example 4. Let ( ) 423 23 −+−= xxxxf , find 1. ( )2−f 2. ( ) ( )12 −− ff 2.6 Exercise Find the domain and range of each relation. 1. {(−9, 12), (0, 8), (12, −15), (3, 15)} 2. {(7, −7), (4, − 7), (−5, 17)} Determine which relations are also functions. 3. {(−9, 12), (0, 8), (12, −15), (3, 15)} 4. {(−2, 4), (12, 14), (−2, − 4)} Use vertical line test to determine whether each graph is the graph of a function. 5.
6.
37
Given the function ( ) xxf 94 −= , find the indicated function values. 7. ( )1−f 8. ( )2f 9. ( )0f Given the function ( ) 72 −= xxf , find the indicated function values. 10. ( )0f 11. ( )5−f 12. ( )2f
38
2.7 Graphing Linear Inequalities in Two Variables Learning Objectives:
1. Determine whether an ordered pair is a solution of a linear inequality in two variables. 2. Graph a linear inequality in two variables. 3. Key Vocabulary: linear inequality in two variables, halfplanes, boundary line.
Linear Inequality—is an equation of the form: CByAx <+ ; CByAx >+ ;
CByAx ≤+ ; CByAx ≥+ A. Graphing Linear Inequalities in Two Variables Steps to graph linear inequalities.
1. Solve inequality in the form .bmxy +> 2. If inequality involving ≥≤ or , draws a solid line. If inequality involving < or >, draws a dashed line. 3. Pick a test point. Substitute the values in the inequality. If the result is true, shade the side that contain the test point. If a false statement, shade the other side.
CAUTION! If multiply or divide by a negative number, the inequality sign change to opposite. Example 1. Graph the following inequality and label at least two points on the graph grid. 1. 43 +−≤ xy 2. 02 ≥+x
y
x
5
55−
5−
y
x
5
55−
5−
39
3. 02 <−y 2.7 Exercise Determine which ordered pairs are solutions of the linear inequality 243 ≥− yx 1. (−1, −2) 2. (2, 1) 3. (3, 2) Graph each inequality and label at least two points on the graph grid. 4. 44 >− yx 5. xy 5< 6. yx 4−≤ 7. 1628 ≥− yx
y
x
5
55−
5−
40
Chapter 3 Systems of Linear Equations 3.1 Solving Systems of Equations by Graphing Learning Objectives:
1. Decide whether an ordered pair is a solution of a system of linear equations. 2. Solve a system of linear equations by graphing. 3. Identify special systems: those with no solution and those with an infinite number of
solutions. 4. Key Vocabulary: system of linear equations, parallel lines, no solution, infinite number of
solutions, inconsistent system, consistent system, dependent equation, independent equations.
A. Deciding Whether an Ordered Pair Is a Solution System of Equations—consists at least two or more linear equations.
Example. ⎩⎨⎧
=−−=−
021234
.1yx
yx
⎪⎩
⎪⎨
⎧
=+=−+=+−
0222
5.2
zxzxy
zyx
Solution of the system—is the point(s) where the graphs intersect. Example 1. Is ( )9,3 a solution of?
⎩⎨⎧
=−=−
xyyx
3325
Three types of the System of Equations. 1. Consistent System
• Two lines intersect at one point ( )yx, . • Has one solution ( )yx, . • 21 mm ≠ • When solve the system, get x = a number, y = a number.
2. Inconsistent System
• Two lines are parallel. • Has no solution. • 21 mm = and 21 bb ≠ • When solve, get false statement.
x
y
x
y
41
3. Dependent System • Two lines lie on top of the others (same line). • Has infinitely many solutions. • 21 mm = and 21 bb = • When solve the system, get true statement.
B. Solving Systems of Equations by Graphing Steps for solving linear system by graphing.
1. Solve and graph each equation separately. 2. Identify type of systems (consistent, inconsistent, or dependent). 3. State number of solution (one solution, infinitely many solutions or no solution).
Example 1. Solve, graph, label type of system and state number of solution.
1. ⎩⎨⎧
=−=+
04242
yxyx
2. ⎩⎨⎧
−=−=−
86233
xyxy
y
x
5
55−
5−
y
x
5
55−
5−
x
y
42
3. ⎩⎨⎧
=+=+
82442
xyyx
3.1 Exercise Determine whether the ordered pair satisfies the system of linear equations.
1. ⎩⎨⎧
−=+−=
152
yxyx
( )1,2− 2. ⎩⎨⎧
=+=−102
632yxyx
( )2,0 −
3.
⎪⎪⎩
⎪⎪⎨
⎧
−=
−=+
xy
yx
31
619
32
43
( )2,6−
Solve each system of equations by graphing. State the solution(s) and type of system.
4. ⎪⎩
⎪⎨⎧
+−=
−=
4
132
xy
xy 5.
3632
==+
xyx
6. ⎩⎨⎧
−==−
413
yyx
7. ⎩⎨⎧
−=−−=+
823423
yxyx
8. ⎩⎨⎧
−=+=−
33413
yxyx
9. ⎪⎩
⎪⎨⎧
−=
=−
121
22
yy
yx
y
x
5
55−
5−
43
3.2 Solving Systems of Linear Equations by Substitution Learning Objectives:
1. Use the substitution method to solve a system of linear equations. 2. Key Vocabulary: substitution method.
Steps to solve linear system by substitution:
1. Solve one of the equations for one of its variable: x or y. 2. Substitute the resulting found in step 1 into the other equation. 3. Solve the equation found in step 2 to find the value of one variable. 4. Substitute the value found step 3 in any original equations containing both variables to find the value of the other variable. 5. Check the solution by substituting the numerical values of the variables in both original equations.
Example 1. Solve, label type of system and state number of solution.
1. ⎩⎨⎧
−=−=−
122663
xyyx
2.
⎪⎪⎩
⎪⎪⎨
⎧
=+
−=+
41
64
13
2
yx
yx
44
3.2 Exercise Solve each system of equations by substitution. State the solution(s) and type of system.
1. ⎩⎨⎧
=+−=
14213
yxyx
2. ⎩⎨⎧
−==−
3274
xyyx
3. ⎩⎨⎧
−=−=+
44366
yxyx
4. ⎪⎩
⎪⎨⎧
=−
=+
8
521
21
yx
yx 5.
⎩⎨⎧
=−−=−
221135
yxyx
6. ⎩⎨⎧
=−=+
yxxy6214
713
7. ⎩⎨⎧
−=−−=−
27472
yxyx
8. ⎩⎨⎧
=−=+64733
yxyx
10. ⎪⎩
⎪⎨⎧
−=
=−
353
1553
xy
yx
45
3.3 Solving Systems of Linear Equations by Addition Method Learning Objectives:
1. Use the addition method to solve a system of linear equations. 2. Key Vocabulary: addition method, elimination method, opposite.
Steps to solve a system of two linear equations by the addition method:
1. Rewrite each equation in standard form: CByAx =+ . 2. If necessary, multiply one or both equations by a nonzero number so that the coefficients of a chosen variable in the system are opposites. 3. Add both equations. 4. Find the value of one variable by solving the resulting equation from step 3. 5. Find the value of the second variable by substituting the value found in step 4 into either one of the original equations. 6. Check the solution by substituting the numerical values of the variables in both original equations.
Example 1. Solve, label type of system and state number of solution.
1. ⎩⎨⎧
−==+
1143645
xyyx
2. ⎩⎨⎧
+−==+
12.002.004.06.01.02.0
yxyx
46
3.
⎪⎪⎩
⎪⎪⎨
⎧
−=+
=+
235
2
565
yx
yx
3.3 Exercise Solve each system of equations by addition. State the solution(s) and type of system.
1. ⎩⎨⎧
−=+−=+
95232
yxyx
2. ⎩⎨⎧
−=−=+
2552053
yxyx
3. ⎩⎨⎧
=−−=−−
183464
yxyx
4. ⎩⎨⎧
+−=−+=
yxyx
4536412
5. ⎩⎨⎧
=+=−
24841863
baba
6. ⎩⎨⎧
=−−=+−
10521687
yxyx
7.
⎪⎪⎩
⎪⎪⎨
⎧
=−
−=+
121
61
421
31
yx
yx 8.
⎪⎩
⎪⎨⎧
+−=
=+
25
21
52
xy
yx 9.
⎪⎩
⎪⎨⎧
+−=
=+
421
82
xy
yx
47
3.4 Applications of System of Linear Equations Learning Objectives:
1. Use a system of equations to solve problems. ProblemSolving Steps: 1. UNDERSTAND the problem by do the following:
• Read and reread the problem. • Identify what is given and what is the question. • Choose two variables to represent the two unknowns being asked. • Construct a drawing if needed.
2. TRANSLATE the problem into two equations. 3. SOLVE the system of equations. 4. INTERPRET the results: Check the proposed solution in the stated problem and state your conclusion.
A. Finding Unknown Numbers Example 1. The sum of two numbers is 56. Their difference is 12. What are the numbers? B. Solving a Problem about Prices Formula: Number of tickets ×price per ticket = total price Example 2. Admission prices at a local weekend fair were $ 5 for children and $7 for adults.
The total money collected was $3379, and 587 people attended the fair. How many children and how many adults attended the fair?
Numbers of tickets × Price per ticket = Total price children × = adults × =
48
C. Coin Problems Total Value = numbers of coins× value of each coin Example 3. Tim has $ 1.10 in quarters and nickels. How many quarters and nickels does she
have if he has 14 coins in total? Numbers of coins × Value of each coin = Total value quarters × = nickels × = D. Investment Problems tI Pr= Where I = interest earn, P = principal, r = interest rate, t = time (in year) Example 4. Lit invested $6000, part at 6% and the rest at 4%. How much is invested at each
rate if the annual income from the two investments is $290?
P × r ×
t = I
Account 1 × ×
=
Account 2 × ×
=
49
E. Mixture Problems Formula: Amount of solution = number of liters × percent of the solution Example 5. A pharmacist wants to make 50 liters of a 60% alcohol solution. She currently has a
20% alcohol solution and a 70% alcohol solution. How many liters of a 20% alcohol solution and a 70% alcohol solution she needs to make 50 liters of a 60% alcohol solution?
Number of liters × Percent of solution = Amount of solution Solution 1 × = Solution 2 × = Mixture × = F. Geometry Problems Example 6. The perimeter of a rectangle is 58 inches. The length is 5 more than three times the width. Find the length and the width.
50
3.4 Exercise Solve each problem using systems of equations. 1. The sum of two numbers is 56. Their difference is 4. Find the two numbers. 2. The sum of two numbers is 44. The second number is 5 more than twice the first. Find the numbers. 3. The difference between two numbers is 16. Five times the smaller is the same as 8 less than twice the larger. Find the numbers. 4. Two records and three tapes cost $31. Three records and two tapes cost $29. Find the cost of each record and tape. 5. At school, two photography packages are available. Package A contains 1 class picture and 10 wallet‐size pictures for $19. Package B contains 2 class pictures and 15 wallet‐size pictures for $31. Find the cost of a class picture and the cost of a wallet‐size picture. 6. A broker invested a total of $4500 in two different stocks. One stock earned 9% per year. The other earned 6% per year. If $360 was earned from the investment, how much money was invested in each? 7. The price of admission for a concert was $9 for adults and $4 for children. Altogether, 1770 tickets were sold, and the resulting revenue was $14,680. How many adults and how many children attended the concert? 8. A druggist has one solution that is 10% iodine and another solution that is 50% iodine. How much of each solution should the druggist use to get 100 ml of a mixture that is 20% iodine? 9. A chemist has one solution that is 20% alcohol and another that is 60% alcohol. How much of each solution should the chemist use to get 100 ml of a solution that is 52% alcohol? 10. The perimeter of a rectangle is 54 cm. Two times the height is 3 cm more than the base. Find the length of the height and length of the base. 11. The sum of the legs of a right triangle is 17 inches. The longer leg is 2 more than twice the shorter. The hypotenuse is 13 in. Find the length of each leg. 12. Two angles are complementary. The larger angle is 6 less than 5 times the smaller angle. Find the measure of each angle. 13. Todd has 27 total coins in his bank, all dimes and quarters. The coins have a total value of $4.95. How many of each coin does he have? 14. Nary has $2.20 in dimes and nickels. She has 26 coins in total. How many dimes and nickels does she have?
51
Chapter 4 Exponents and Polynomials
4.1 EXPONENTS Learning Objectives:
1. Evaluate exponential expressions. 2. Use the product rule for exponents. 3. Use the power rule for exponents, products, and quotients. 4. Use the quotient rule for exponents, and define a number raised to the 0 power. 5. Decide which rule(s) to use to simplify an expression. 6. Key Vocabulary: exponential expression, power, raised, product rule, same base, simplifying
an exponential expression, power rules, quotient rule, zero exponent. Exponential Expression—is expression of the form: 43421
timesn
n aaaaa ⋅⋅⋅⋅⋅⋅= , where a is the based, n is the exponent.
Exponential Properties. If m and n are integers, and x and y are any real number, 0y,0x ≠≠ , then
1. =⋅ nm xx 2. ( ) =nmx 3. ( ) =mxy
4. =⎟⎟⎠
⎞⎜⎜⎝
⎛m
yx 5. =
n
m
xx 6. =0x
7. =− 0x 8. ( ) =− 0x CAUTION! ( ) mmm yxyx +≠+ and ( ) mmm yxyx −≠− Example 1. Evaluate each expression.
1. ( )27− 2. 26− 3. 2
41⎟⎠⎞
⎜⎝⎛−
52
4. 243 ⋅ 5. 02 88 − 6. 2y4− when 5−=y Example 2. Use the properties of the exponent to simplify. Write the results using exponents. 1. ( )( )6423 yx5yx3 −−
2. 2
3
42
z2yx3
⎟⎟⎠
⎞⎜⎜⎝
⎛−
3. ( )( )222
4
ba6
ab12
4. ( )052 yx5
53
5. 00 9x +
4.1 Exercise
Evaluate each expression.
1. 28 2. 28− 3. ( )28− 4. 2
71⎟⎠⎞
⎜⎝⎛−
5. 25 22 − 6. 6x2 when x = −2 7. 3xy
2 when x = − 1 and y = −5
Simplify each expression.
8. (4y3)(−3y
7) 9. (3z
11)(−5z
2)(z
4) 10.
2
2
33⎟⎟⎠
⎞⎜⎜⎝
⎛−
zxy 11. 33
73
248
yxyx
12. −3x0
13. ( )32310 zxy− 14. 3
3
7
204
⎟⎟⎠
⎞⎜⎜⎝
⎛xy
54
4.2 Negative Exponents and Scientific Notation Learning Objectives:
1. Simplify expressions containing negative exponents. 2. Use the rules and definitions for exponents to simplify exponential expressions. 3. Convert numbers in standard form to scientific notation. 4. Convert numbers in scientific notation to standard form. 5. Key Vocabulary: negative exponents, scientific notation, standard form.
Properties of Negative Exponents: If m and n are integers, and x and y are any real number, 0y,0x ≠≠ , then
1. n
n
x1
x =− 2. nn
xx1
=−
3. =⎟⎟⎠
⎞⎜⎜⎝
⎛−m
yx
m
mm
xy
xy
=⎟⎠⎞
⎜⎝⎛
4. =⋅ nm xx nmx + 5. ( ) =nmx mnx 6. ( ) =mxy mm yx
7. =⎟⎟⎠
⎞⎜⎜⎝
⎛m
yx
m
m
yx 8. =
n
m
xx nmx − =
mnx1− 9. =0x 1
10. 10 −=− x 11. ( ) 10 =− x Example 1. Write using positive exponents and simplify the following.
1. 33− 2. 3
21
−
⎟⎠⎞
⎜⎝⎛ 3.
4
a2
−
⎟⎠⎞
⎜⎝⎛
4. 21 55 −− + Example 2. Write using negative exponents.
1. 581 2.
6x7
55
Example 3. Performed the indicated operation and simplify. Write answer using positive exponents.
1. 64 33 ⋅− 2. 77 xx ⋅−
3. 6
4
yy −
4. 3
36
45
ba4ba3
−
− ⎟⎟⎠
⎞⎜⎜⎝
⎛
Scientific Notation—is an expression of the form: n10a× where 10a1 <≤ and n is the power. 1. Convert a number to scientific notation Steps: 1. If decimal point in the given number moves to left, n is positive.
2. If decimal point in the given number moves to right, n is negative. Example 4. Write the given number in scientific notation. 1. 6,350,000 2. 0.00245 2. Convert scientific notation to a number Steps: 1. If n is positive, moves decimal point in the given number to the right. 2. If n is negative, move decimal point in the given number to left.
56
Example 5. Write the given scientific notation in standard notation. 1. 41056.6 × 2. 3104.3 −× 4.2 Exercise Simplify each expression. Write results with positive exponents.
1. 25− 2. 4
31 −
⎟⎠⎞
⎜⎝⎛− 3. 11 54 −− + 4. 7
6
−
−
qp
5. ( ) 343 −yx 6. 2
92
35
⎟⎟⎠
⎞⎜⎜⎝
⎛ −
yxyx 7. ( )
( ) 12
345−−
−−−
xyxy 8. ( )
( ) 21
432
3 −− bacba
Write each number in scientific notation. 9. 12,000,000,000 10. 0.00031 Write each number in standard form. 11. 710784.1 −× 12. 1010052.6 ×
57
4.3 Introduction to Polynomial Learning Objectives:
1. Define term and coefficient of a term. 2. Define polynomial, monomial, binomial, trinomial, and degree. 3. Evaluate polynomials for given replacement values. 4. Simplify a polynomial by combining like terms. 5. Simplify a polynomial in several variables. 6. Write a polynomial in descending powers of the variable and with no missing powers of
the variable. 7. Key Vocabulary: coefficient, constant, polynomial, monomial, binomial, trinomial, degree of
a term, degree of the polynomial. A. Classifying Polynomial Polynomial—is a finite sum of terms of the form nax , where a is a real number and n is a whole
number. Term—is a number or the product of a number and variables raised to powers separated by plus
or minus signs. Numerical Coefficient (coefficient)—is the numerical factor of each term. Constant term—is the term that contains only a number. Types of Polynomials 1. Monomial—is a polynomial with one term. 2. Binomial— is a polynomial with two terms. 3. Trinomial— is a polynomial with three terms. 4. Polynomial— is a polynomial with four or more terms. Example 1. Classify as monomial, binomial, or trinomial. 1. 2643 yy ++− 2. )1(3)9(8 −−+ xx B. Finding the Degree of a Polynomial Degree of a Polynomial—is the greatest degree of any term of the polynomial. Note: 1. A constant term has zero degree. 2. Zero polynomial has no degree. Example 2. Find the degree of the terms and the degree of the polynomial. 1. 9 2. 825 2 −+− zz 3. 42222 yxyxyx +−
58
C. Writing a Polynomial in Descending Order Example 3. Write in descending order: 1. xxx 2834 32 +−+− 2. 13 2 −+− yy D. Evaluating Polynomials Example 4. Find the value of ,9016)( 2 +−= ttP when 2=t E. Simplifying Polynomials by Combining Like Terms Like Terms—are terms that contain exactly the same variables raised to exactly the same powers. To Combine Like Terms—is to combine the coefficient of the like term. Example 5. Simplify by combining like terms. 1. 3223 5263 yxxy −+− 2. 2.88.16.113.67.3 33 −−++− xxxx
3. xxxx83
212
41
72 33 +−+−
59
4. 22222222 5427 xyxxyyyx +−+−+ Example 5. Find the total area of the rectangles. 4.3 Exercise Simplify each expression. Write results with positive exponents. 1. Complete the table for the polynomial 745 23 −−− xxx
Term Coefficient 35x
4−
x
7−
Find (a) the degree of each of each term (b) the degree of the following polynomials (c) determine whether it is a monomial, binomial, trinomial, or none of these. 2. 127 −x 3. 2435 xx +− 4. 43342 537 ababa ++−
x
2 x2
x3 x2
x2
6
3
x4
x2 3
1
60
Find the value of each polynomial when (a) 0=x and (b) 1−=x 5. 43 −x 6. 262 +− xx 7. 725 23 +− xx Simplify each of the following by combining like terms. 8. 33 912 xx + 9. xxx 6177 22 +− 10. 22 4.31.36.24.5 yyyy −−−
61
4.4 Adding and Subtracting Polynomials Learning Objectives:
1. Add polynomials. 2. Subtract polynomials. 3. Add or subtract polynomials in one variable. 4. Add or subtract polynomials in several variables. 5. Key Vocabulary: combine like terms.
A. Adding or subtracting Polynomials To Add or to subtract Polynomials is to add or subtract the coefficient of the like terms. Example 1. Perform the indicated operation 1. Add: 385 2 −+ xx and xx 583 2 −+− 2. Subtract 2845 yy +− from yy 46 2 − 3. Subtract 13 +x from the sum of 34 −x and 25 +x . 4. ( ) ( )2222 7362 babababa −+−++− 5. ( ) ( )22222222 287935 yxyyxyyxyx +−+−−+−+
62
4.4 Exercise Add or subtract as indicated. 1. ( ) ( )95374 2 +−+− xxx 2. ( ) ( )19217312 22 −+++− aaaa 3. ( ) ( )45753 22 −−−+− aaa 4. ( ) ( ) ( )83467415 222 −−−−++− aaaaa
5. ⎟⎠⎞
⎜⎝⎛ −+−⎟
⎠⎞
⎜⎝⎛ −+ 2
65
948
43
31 2222 yxyx 6. ( ) ( )222222 5117772 yxyxyxxyyx +−−−+−
63
4.5 Multiplying Polynomials Learning Objectives
1. Multiply monomials. 2. Multiply a monomial by a polynomial. 3. Multiply two polynomials. 4. Multiply polynomials vertically. 5. Key Vocabulary: polynomial, monomial, binomial, trinomial.
A. Multiplying Two Monomials Steps. 1. Multiply the coefficient with the coefficient. 2. Multiply the like variable by adding their power. Example 1. Multiply: 1. ( )( )43 54 yy− 2. ( )( )xx −− 45 B. Multiplying a Monomial and a Binomial Distributive Property: ( ) acabcba +=+ and ( ) acabcba −−=+− Example 2. Multiply: 1. ( )ba 34 − 2. ( )4332 23 −+− xxx C. Multiplying Two Binomials Using Distributive Property
( )( ) =++ dcba ( ) ( ) bdbcadacdcbdca +++=+++ Example 3. Multiply.
64
1. ( )( )1432 −− aa 2. ( )252 yx − 3. ( )( )4523 2 +−− xxx 4.5 Exercise
1. ( )( )43 32.4 xx− 2. ⎟⎠⎞
⎜⎝⎛−⎟⎠⎞
⎜⎝⎛− 56
103
95 yy 3. ( )( )( )72 245 xxx −
4. ( )5353 2 −−− baba 5. ( )32232 9745 bbaaab −− 6. ( )( )5247 +− xx
7. ⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ +
41
43 xx 8. ( )( )343 2 +−− yyy 10. ( )( )52552 22 −++− xxxx
11. Find the area of a rectangle with sides ( )34 +x ft. and ( )72 −x ft. 12. Find the area of a triangle with height x6 in. and base ( )19 −x in.
65
4.6 Special Products of Polynomials Learning Objectives:
1. Multiply two binomials using the FOIL Method. 2. Square a binomial. 3. Multiply the sum and difference of two terms. 4. Use special products to multiply binomials. 5. Key Vocabulary: FOIL method, squaring a binomial.
Special Products of Polynomials: 1. Product of Two Binomials: ( )( ) bcbdadacdcba +++=++ 2. The Square of a Binomial Sum: ( ) ( )( ) 222 2 bababababa ++=++=+ 3. The Square of a Binomial Difference: ( ) ( )( ) 222 2 bababababa +−=−−=−
4. The Product of the Sum and Difference of Two Terms: ( )( ) 22 bababa −=−+ Example 1. Multiply. 1. ( )( )5392 −+ yy 2. ( )232 −y 3. ( )32+y
4. ⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ + yxyx
312
312
66
5. ( )( )3232 ++ yyy 4.6 Exercise Multiply.
1. ( )210+x 2. ( )2211 −a 3. 2
75⎟⎠⎞
⎜⎝⎛ −y 4. ( )( )1010 −+ xx
5. ( )( )4747 22 +− yy 6. ( )( )yxyx −+ 88 7. ⎟⎠⎞
⎜⎝⎛ +⎟⎠⎞
⎜⎝⎛ − yaya
53
53
67
4.7 Division of Polynomials Learning Objectives:
1. Divide a polynomial by a monomial. 2. Use long division to divide a polynomial by a polynomial other than a monomial. 3. Key Vocabulary: dividend, quotient, divisor.
A. Dividing a Polynomial by a Monomial
Addition Rule: 0; ≠+=+ C
CB
CA
CBA
Example 1. Divide the following.
1. 2
23
41248
yyyy +−
2. 234 927 xxx −− by 29x− B. Dividing a Polynomial by a Binomial Example 2. Divide 162 2 −+ yy by 5+y
68
4.7 Exercise Divide.
1. 6
182436 2 −− xx 2. p
pp7
2142 23 − 3. a
xx4
61216 3
−+−
4. xy
yxxyyx33327 2432 −− 5.
14117 2
−+−
xxx 6.
52021203 23
+−++
aaax
7. 52
1024236 23
−−+−
xxxx
69
Chapter 5 Factoring Polynomials 5.1 The Greatest Common Factors Learning Objectives:
1. Find the greatest common factor of a list of numbers. 2. Find the greatest common factor of a list of terms 3. Factor out the greatest common factor from the terms of a polynomial. 4. Factor by grouping. 5. Key Vocabulary: factors, factored form, factoring, factoring out, greatest common factor,
GCF, factoring by grouping. A. Finding the Greatest common Factor (GCF) of Numbers Greatest common Factor (GCF)—is the largest common factor of the integers in the list. Steps to find the GCF. 1. Write each of the numbers as a product of prime number using exponent for repeated number. 2. Choose the number that has the lowest exponent, then find their product. 3. For the common variable, choose the smallest exponent Example 1. Find the GCF of: 1. 45, 60 and 108 2. 58 9,12 xx and 830x−
70
B. Factoring Out the Greatest Common Factor Example 2. Factor completely: 1. 186 +a 2. 42 366 xx −− 3. 3567 25102015 xxxx ++−
4. 52
51
54 2 +− aa
5. ( ) ( )225 −−− yxy 5.1 Exercise Find the GCF for each list. 1. 35, 45 2. 748 ,, aaa 3. 22324 28,7,14 yxyxyx− Factor out the GCF from each polynomial. 4. 124 −a 5. xyyxyx 2763 223 ++ 6. 2131563 234 +−++ yyyy 7. 22355 21147 yxyxyx −− 8. ( ) ( )323 +++ xxy 9. ( ) ( )232 22 +−+ bba Factor the following four‐term polynomials by grouping. 10. yxyx 5153 +++ 11. yxxyx 217124 2 +−− 12. yxxyx 10563 2 +−− 13. 35202112 23 −−+ xxx
71
5.2 Factoring Trinomial of the Form cbxx ++2 Learning Objectives:
1. Factor trinomials of the form cbxx ++2 . 2. Factor out the greatest common factor and then factor a trinomial of the form cbxx ++2 . 3. Key Vocabulary: factor, product, sum, binomials, prime.
A. Factoring Trinomials of the Form cbxx ++2 Steps for Factoring cbxx ++2 : Find two integers whose product is c and whose sum is b. 1. If b and c are positive, then both integers must be positive.
cbxx ++2 = ( )( )++ xx 2. If c is positive and b is negative, then both integers must be negative.
cbxx +−2 = ( )( )−− xx
3. If c is negative, one integer must be positive and one negative. cbxx −+2 = ( )( )−+ xx bigger number is +; smaller number −is cbxx −−2 = ( )( )−+ xx smaller number is +; bigger number −is
Example 1. Factor completely: 1. 782 +− xx 2. 298 xx ++ 3. 1442 −+ yy 4. 425262 20 ybybyb ++−
72
5. 22 86 aaxx ++ 5.2 Exercise Factor the following trinomials completely. Write “prime” if they do not factor. 1. 36132 ++ xx 2. 56152 +− xx 3. 2762 −− xx
4. 10832 −+ xx 5. 372 ++ xx 6. 21102 +− xx
7. 2452 −− xx 8. 6322 −+ xx 9. 60355 2 ++ xx
10. 2336105 xx +− 11. xxx 54 23 −− 12. 32 214 yyy +−−
13. 234 32202 yyy +− 14. xxx 105363 23 +− 15. 534 3549 yyy +−−
73
5.3 FACTORING TRINOMIALS OF THE FORM 1,2 ≠++ acbxax Learning Objectives
1. Factor trinomials of the form 1,2 ≠++ acbxax . 2. Factor out the GCF before factoring a trinomial of the form cbxax ++2 . 3. Key Vocabulary: greatest common factor, coefficient.
A. The ac Test for cbxax ++2 ac TEST for Factoring of cbxax ++2
cbxax ++2 is factorable if there is two integers with product ac and sum is b. Example 1. Determine whether the polynomial is factorable: 1. 234 2 ++ yy 2. 253 2 ++ yy 3. xx 329 2 −− B. Factoring cbxax ++2 by trial and error Example 2. Factor completely: 1. 253 2 ++ xx 2. 6318 2 −+ xx
74
3. 22 656 yxyx −− 4. yyy 21112 23 −−− 5.3 Exercise Complete the following. 1. ( )( )1316 2 +=−− yyy 2. ( )( )52584 2 −=−− yyy 3. ( )( )2310133 2 +=−− yyy 4. ( )( )25215 2 +=−+ yyy Factor the following trinomials completely. Write “prime” if they do not factor. 5. 2715 2 −+ xx 6. 3108 2 +− xx 7. 10116 2 −− xx 8. 8189 2 ++ xx 9. 21721 2 +− xx 10. 10318 2 −− aa 11. 151112 2 −− xx 12. 4415 2 −− aa 13. 144514 2 −− aa 14. 2444 2 ++− aa 15. 359142 2 ++ xx 16. 22 156550 yxyx −+ 17. xxx 1053318 23 −− 18. aaa 256530 23 +− 19. 4322 31310 yxyyx −−
75
5.4 Factoring Trinomials of the Form 1,2 ≠++ acbxax by Grouping
Learning Objectives 1. Use the grouping method to factor trinomials of the form ax2 + bx + c. 2. Key Vocabulary: grouping method, trinomial.
A. Factor using the Grouping Method Step for factoring a trinomial by grouping
1. Factor out a greatest common factor, if there is one other than 1. 2. For the resulting trinomial 1,2 ≠++ acbxax , find two numbers whose product is ac and whose sum is b. 3. Write the middle term, bx, using the factors found in step 2. 4. Factor by grouping.
Example 1. Factor completely. 1. 9124 2 +− xx 2. 501520 2 −− xx 3. 2615 xx +− 4. 6197 2 −− xx
76
5. xxx 504545 23 −+ 5.4 Exercise Factor by grouping. 1. 231421 2 +++ xxx 2. 5153 2 +−− xxx 3. 3186 2 +++ xxx 4. 273 2 ++ xx 5. 295 2 −− xx 6. 31710 2 ++ nn 7. 18174 2 +− mm 8. 6313063 2 ++ aa 9. aaa 62212 23 ++ 10. mmm 64260 23 +− 11. xxx 9811218 23 −+ 12. xxx 356510 23 −+ 13. yxyyx 5136 2 −+
77
5.5 Factoring Squares of Binomials Learning Objectives:
1. Recognize perfect square trinomials. 2. Factor perfect square trinomials. 3. Factor the difference of two squares. 4. Key Vocabulary: perfect squares, perfect square trinomial, square of a binomial, difference
of two squares. Factoring Perfect Square Trinomials 1. Perfect Square Trinomials. ( )( ) ( )222 2 bababababa +=++=++ ( )( ) ( )222 2 bababababa −=−−=+− 2. Difference of Two Squares. ( )( )bababa −+=− 22 22 ba + is prime (cannot be factored) To be a perfect square trinomial, a trinomial must satisfy 3 conditions:
1. The first and last terms ( 2a and 2b ) must be perfect squares 2. There must be no minus signs before 2a and 2b 3. The middle term is ab2 or ab2−
Example 1. Determine whether the expression is the perfect square: 1. 22 4129 yxyx +− 2. 992 ++ yy Example 2. Factor completely: 1. 962 ++ yy
79
5.5 Exercise 1. Is 11025 2 ++ yy a perfect square trinomial?
Fill in the blank so that the trinomial is a perfect square trinomial 2. 25_______16 2 ++ xx 3. 121_______9 2 ++ yy Factor completely. 1. 49142 ++ xx 2. 25309 2 +− xx 3. 9622 ++ xyyx
4. 456020 2 ++ xx 5. aaa 147423 23 ++ 6. 22 506018 ayaxyax +−
7. 252 −x 8. 499 2 −x 9. 264 y−
10. 2251 x− 11. 494 −x 12. 8116 4 −y
13. yy 218 3 − 14. 24 1275 xx − 15. 6492 −x
80
5.6 Solving Quadratic Equations by Factoring Learning Objectives:
1. Solve quadratic equations by factoring. 2. Solve equations with degree greater than 2 by factoring. 3. Key Vocabulary: quadratic equation, standard form, zero factor property, GCF.
Quadratic Equation in Standard Form—is an equation of the form
02 =++ cbxax , where a, b, c are real numbers. Principle of Zero Product: If 0or00 === bathenab Steps for Solving Quadratic Equations
1. Perform the necessary operations on both sides of the equation so that the right‐hand side is 0. 2. Use the general factoring strategy to factor the left side of the equation, if necessary. 3. Use the Principle of Zero Product to solve each of the resulting equations. 4. Check the results by substituting the solutions obtained in step 3 in the original equation.
Example 1. Solve each of the following 1. ( )( ) 0431059 2 =−−+ xxx 2. 383 2 =+ xx
81
3. ( ) 132 −=+yy 4. ( )( ) ( ) 713225 −+=+− xxx 5.6 Exercise Solve each equation. 1. ( )( ) 0211 =−+ xx 2. ( )( ) 01325 =+− xx 3. ( )( ) 02152 =+− xxx
4. ( )( ) 05275 =−− xx 5. 0652 =−+ xx 6. 0532 2 =−− xx
7. 0254 2 =−y 8. 025204 2 =++ xx 9. mm 112 =
10. xx 4322 −= 11. 169 2 =y 12. 0128 23 =++ nnn
13. yy 28 3 = 14. 24235 2 −=− pp 15. www 80405 23 −=
82
5.7 Quadratic Equations and Problem Solving Learning Objectives:
1. Translate words to algebraic expressions. 2. Solve problems that can be modeled by quadratic equations. 3. Key Vocabulary: Pythagorean Theorem, right triangle, hypotenuse, legs.
Consecutive Numbers 1. Consecutive
Number 1st number
x 2nd number
1+x 3rd number
2+x 4th number
3+x 2. Consecutive Odd Number
1st number x
2nd number 2+x
3rd number 4+x
4th number 6+x
3. Consecutive Even Number
1st number x
2nd number 2+x
3rd number 4+x
4th number 6+x
Example 1. The square of a number minus twice the number is 63. Find the numbers. Example 2. The length of a rectangular garden is 5 feet more than its width. The area of the
garden is 176 square feet. Find the length and the width. Example 3. Find two consecutive odd integers whose product is 23 more that their sum.
83
Example 4. A student dropped a ball from the top of a 64‐foot building. The height of the ball after t seconds is given by the quadratic equation h = −16t
2 + 64. How long will it
take the ball to hit the ground? Pythagorean Theorem In any right triangle 222 cba =+ Example 5. The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 m. Find the lengths of the legs.
Leg b
Leg a Hypotenuse c
84
5.7 Exercise Solve. 1. A rectangle has an area of 24 square inches. The width is represented by 3−x and the length is 2+x . Find the dimensions. 2. The length of a rectangle is 3 cm more than the width. The area is 70 2cm . Find the dimensions of the rectangle. 3. The length of a proposed rectangular flower garden is 6 m more that its width. The area of the proposed garden is 72m2. Find the dimensions of the proposed flower garden. 4. A square field had 5 m added to its length and 2 m added to its width. The field then had an area of 130 m2. Find the length of a side of the original field. 5. A rock is dropped from a 784 foot cliff. The height h of the rock after t seconds is given by the equation h = −16t
2 + 784. Find out how many seconds pass before the rock reaches the
ground. 6. One leg of a right triangle measures 6 m while the length of the other leg measures x meters. The hypotenuse measures (2x – 6) m. Find the length of all 3 sides. 7. The longer leg of a right triangle measures two feet more than twice the length of the shorter leg. The hypotenuse measures 3 feet more than twice the shorter leg. Find the length of all three sides. 8. Find the length of a ladder leaning against a building if the top of the ladder touches the building at a height of 12 feet. Also, the length of the ladder is 4 feet more than its distance from the base of the building. 9. One leg of a right triangle is 14 meters longer than the other leg. The hypotenuse is 26 meters long. Find the length of each leg. 10. Eight more than the square of a number is the same as six times the number. Find the number. 11. Fifteen less than the square of a number is the same as twice the number. Find the numbers. 12. Seven less than 4 times the square of a number is 18. Find the number. 13. Find two consecutive positive odd integers whose product is 35. 14. The sum of the squares of two consecutive integers is 41. Find the integers. 15. Find two consecutive odd integers such that the square of the first added to 3 times the second is 24.
85
Chapter 6 Rational Expressions and Functions 6.1 Simplifying Rational Expressions Learning Objectives:
1. Find the value of a rational expression given a replacement number. 2. Identify values for which a rational expression is undefined. 3. Simplify or write rational expressions in lowest terms. 4. Write equivalent forms of rational expressions. 5. Key Vocabulary: rational expressions, simplifying rational expressions.
Rational Expression—is an expression of the form QP ; P and Q are any polynomials; 0≠Q
A. Evaluating Rational Expressions Standard Form of a Fraction: For 0≠b
1. ba
ba
ba
ba
−−
−=−
=−
=− 2. ba
ba
ba
ba
=−
−=−
−=−−
Example 1. Find the value of 3;225
2
2
−=−−−+ x
xxxx
B. Identifying When a Rational Expression is Undefined A rational expression is undefined when the denominator is 0. Example 2. Find values for which the rational expression is undefined.
1. 32 +m
m
86
2. 54
12 −−
+yy
y
3. 93
2 ++
nn
C. Simplifying Rational Expressions
1. Fundamental Rule of Fractions: BA
BCAC
= ; 0,0 ≠≠ CB
2. Quotient of Additive Inverses (“1 trick”)
a. 1=++
=++
baab
abba b. 1−=
−−
=−−
baab
abba
Step to Simplify a Rational Expression 1. Completely factor the numerator and denominator. 2. Cancel the common factor in the numerator and denominator by replace the quotient of the
common factors by the number 1, since 1=aa .
3. Rewrite the expression in simplified form. Example 3. Simplify.
1. 4
2
63xy
yx−
2. ( )( )nm
nm−−−−
36 22
88
6.1 Exercise Find the value of the following rational expressions when x = 3 and y = −2
1. 724−+
xx 2.
13
2
−yy 3.
837
2
2
−−++
yyyy
Find any real number for which each rational expression is undefined.
4. x
x7
5+ 5. 303175 3
−+
xx 6.
19272 +− xx
Simplify each rational expression.
7. 1812
6−x
8. 205123
−−
xx 9.
xx++
1919
10. x
x−−
1212 11.
baba
+−− 88 12.
2453
2 −−+xx
x
13. 14
5194 2
−−+
xxx 14.
811872
2
2
+−−−xx
xx 15. 366722 2
−−
xx
89
6.2 Multiplication and Division of Rational Expressions Learning Objectives:
1. Multiply rational expressions. 2. Divide rational expressions. 3. Convert between units of measure. 4. Key Vocabulary: reciprocal, unit fractions.
A. Multiplying Rational Expressions
Multiplication Rule: BDAC
DC
BA
=⋅ ; 0,0 ≠≠ DB
Steps for multiplying a rational expression 1. Completely factor numerators and denominators. 2. Multiply the numerators and the denominators. 3. Simplify by canceling the common terms in the numerator and denominators.
Example 1. Simplify and multiply.
1. 22 2115
57
mn
nm⋅
−
2. 22
1075
xyx ⋅
3. 3
22
122
+−
⋅−−−
yy
yyy
4. 1223
25316
2
2
2
2
−−−−
⋅++−+
xxxx
xxxx
90
B. Dividing Rational Expressions
Division Rule: BCAD
CD
BA
DC
BA
=⋅=÷ ; 0,0,0 ≠≠≠ CDB
Example 2. Simplify and divide.
1. ( )3592
−÷+− y
yy
2. y
yyy
−−
÷−+
416
44 2
3. 145103
762
2
2
2
2
−+−−
÷−+−+
xxxx
xxxx
6.2 Exercise Perform the indicated operation and simplify if possible.
1. xx
x47
148 3
⋅ 2. 342
3
549 b
baba
⋅− 3. 5
3217
2 xxx
x −⋅
−
91
4. y
yyyyy 4
3651811
2
2 +⋅
−−+− 5. 22
222
2954
2054
yxyxxx
xyx
−−+
⋅+− 6. 2
2
4
3
2312
yyx
yx
÷
7. ( )( )( ) 83
26834
4−
÷−+
+xxx
x 8. aba
ababa
−÷
+−
2
22 2 9. 124145
62
2
2
−−−−
÷−+
xxxx
xx
10. xyx
yxx
125
825102 −
÷+− 11.
xx
xxxx
xx
−+
÷⎟⎟⎠
⎞⎜⎜⎝
⎛++
−+⋅
−−
613
4133242
3616
2
2
2
2
92
6.3 Adding and Subtracting Rational Expressions Learning Objectives:
1. Add and subtract rational expressions with a common denominator. 2. Find the least common denominator of a list of rational expressions. 3. Write a rational expression as an equivalent expression whose denominator is given. 4. Key Vocabulary: least common denominator (LCD, equivalent expressions.
A. Adding and Subtracting Rational Expressions with the Same Denominator
Rule: B
CABC
BA +
=+. and
BCA
BC
BA −
=−. ; 0≠B
Example 1. Perform the indicated operations.
1. ( ) ( )151
154
−+
− yy
2. ( ) ( )272
279
+−
+ yy
3. 214
52214
2322 −+
−−
−++
xxx
xxx
B. Finding the Least Common Denominator of Rational Expressions Lowest Common Denominator (LCD)—is the smallest algebraic expression that is a multiple of all the denominators. Example 2. Find the LCD for each rational expression.
1. 45 361,
201
xx
93
2. aaa 9
6,273
42 ++
C. Writing Equivalent Rational Expressions
Recall Multiply Rule: BA
BCAC
=
Example 3. Rewrite each rational expression as an equivalent rational expression with the
given denominator.
1. mm 72
?83
=
2. 22 9?
3 babaa
−=
+
3. ( )( )( )531?
3225
23 ++−=
−+−
xxxxxxx
94
6.3 Exercise Add or subtract as indicated. Simplify the result if possible.
1. 1711
17+
a 2. nm
nm
411
413
+ 3. 8
408
5−
−− yyy
4. 61910
6712
−+
−−+
xx
xx 5.
28396
28327
22 −+−
−−+
−xx
xxx
x
Find the LCD for the following lists of rational expressions.
6. yx
y 157,
515
7 7. 159
2,31
−yx
y 8.
xyyx −−31,24
9. 209
1211,3512
922 ++
−++ xx
xxx
Rewrite each rational expression as an equivalent rational expression whose denominator is the given polynomial.
10. ( )1333937
+=
++
abaa 11.
45361512411
22 −=
−−
xxy
12. ( )( )( )44832410
2 −++=
−+ xxxxx 13. ( )( )( )364183
723 +−+
=−−
−xxxxxxx
x
95
6.4 Adding and Subtracting Rational Expressions with Different Denominators Learning Objectives:
1. Add and subtract rational expressions with different denominators. A. Adding and Subtracting Rational Expressions with Different Denominators Steps for Adding or Subtracting Rational Expressions with Different Denominators 1. Find the LCD. 2. Write all fractions as equivalent ones with the LCD as the denominator. 3. Add (or subtract) numerators and keep denominators. 4. Reduce the result if possible. Example 1. Perform each indicated operation. Simplify if possible.
1. y2
53+
2. yy −
−−−
47
45
3. 1
12
2+
−− yy
96
4. 6
29
322 −−
+− xxx
5. 23
265 22 ++−
++ xxxxx
6.4 Exercise Add or subtract as indicated. Simplify the result if possible.
1. xx 53
47+ 2. 23
24xx
+ 3. 63
32
5−
−− xx
4. 16287
92 −
+− x
xx
5. xx −
+− 9
39
11 6. 54+
x 7. 3
710
+−x
8. ( )26
16
1+
−+ xx
9. 2
137
−+
bb 10.
23
1054
+−
−+ a
aa
a 11. 7
656
172 −
−−+ xxx
12. 910
198 22 +−−
−− xxxxx 13.
1003
20122
2084
222 −+
++−
−− xxxxx
97
6.5 Solving Equations Containing Rational Expressions Learning Objectives:
1. Solve equations containing rational expressions. 2. Solve equations containing rational expressions for a specified variable. 3. Key Vocabulary: rational expressions.
A. Solving Rational Equations Steps to solve rational equations.
1. Find the LCD. 2. State the restriction (numbers that make the rational expression undefined). 3. Multiply the numerator of each term by the LCD to clear fraction. 4. Solve the equation and compare the result with the restriction. 5. Verify the answer.
Example 1. Solve and state the restriction of each rational equation.
1. 1031
54
=+x
2. 762 +=+ xx
99
B. Solving Fractional Equations for a Specified Variable
Example 2. Solve for a: ( )1r
1raSn
−−
=
Example 3. Solve for b: xba111
=+
6.5 Exercise Solve each equation.
1. 427
=+x 2.
yy 623
615=+ 3.
4423
−=
−+
aa
a
4. 132
7=
−a 5.
16
1 +=
− xxx 6.
322
32
−+
=−+ x
xx
x
7. 81
82
42
=−
−+ tt 8.
131
192
131
2 −−
−=
+ yyy
Solve each equation for the indicated variable.
9. 2b
hA= for h 10.
2h
bBA
=+
for B 11. GVRN += for V
100
Chapter 7 Roots and Radicals
7.1 Introduction to Radicals Learning Objectives:
1. Finding square roots. 2. Finding cube roots. 3. Finding nth roots. 4. Approximating square roots. 5. Simplifying radicals containing variables. 6. Key Vocabulary: root, positive or principal square root, negative square root, radical,
radical sign, radicand, radical expression, cube root, index, irrational number. A. Finding Square Roots If a is a positive real number, and if ab =2 then ab = and ab −= Square Roots Property: For any real number 0≥a , then 1. aaaa =⋅=2 2. aaaa −=⋅−=− 2 3. a− = not a real number 4. 00 = 5. 11 = Example 1. Find the square root of the following. 1. 169
2. 2581
−
3. 81.0 4. 25−
101
B. Finding the Higher Roots 1. Cube Roots Property: For any real number, then aaaaa =⋅⋅= 33 3
2 Fourth Roots Property: For any real number, 0≥a , then aaaaaa =⋅⋅⋅= 44 4
3 Fifth Roots Property: For any real number, then aaaaaaa =⋅⋅⋅⋅= 55 5 Example 2. Find the root or indicate that the root is not a real number. 1. 3 27
2. 3278
−
3. 4 81− 4. 4 81− Example 3. Find each root. Assume that all variables represent positive numbers. 1. 481x 2. 10 8 2x y z 3. 3 6 9 327a b c
102
7.1 Exercise Find each square root. Indicate if the number is not real.
1. 81 2. 491 3. 144− 4. 9− 5. 3 64−
6. 3 125 7. 4 16− 8. 5 243 9. 5 1024− Find each root. Assume that each variable represents a nonnegative real number. 10. 616y 11. 2081z 12. 3 153125 yx−
103
7.2 Simplify Radicals Learning Objectives:
1. Use the product rule to simplify radicals. 2. Use the quotient rule to simplify radicals. 3. Use both rules to simplify radicals containing variables. 4. Simplify cube roots. 5. Key Vocabulary: perfect squares.
A. The Product Rule and Quotient Rule
1. Product Rule for Radicals: If a and b are two real numbers, then baba ⋅=⋅ 2. Quotient Rule for Radicals:
If a and b are two real numbers, then ba
ba= ; 0≠b
Example 1. Simplify completely. 1. 98 2. 1072x
3. 3 40 4. 167
5. 1083016 6.
246 34ba
104
7. ab
ababab
545
56 22
⋅
7.2 Exercise Simplify each expression. Assume all variables represent real numbers.
1. 75 2. 99 3. 72 4. 6445 5.
2584
−
6. 325y 7. 8645 yx 8. 456y
9. 10
298x
y 10. 20
9
yx
11. 256 12. 67117 yx 13. 3 56 14. 312514 15. 3
12532
105
7.3 Adding and Subtracting Radicals Learning Objectives:
1. Add or subtract like radicals. 2. Simplify square root radical expressions, and then add or subtract any like radicals. 3. Simplify cube root radical expressions, and then add or subtract any like radicals. 4. Key Vocabulary: like radicals.
A. Properties for Adding or Subtracting Radicals 1. ( ) bcabcba +=+ 2. ( ) bcabcba −=− Example 1. Add or subtract as indicated. 1. 20 5 3 5+
2. 92
252
+
3. 3 7 5 21 8 21 10 7+ − − Example 2. Add or subtract by first simplifying each radical and then combining any like
radicals. Assume that all variables represent positive numbers. 1. 10 48 3 75− −
106
2. xx 18685 +− 3. 22 835 xxx ++− 4. 10128123 33 −+−
5. 33 2505128 −
7.3 Exercise Add or subtract as indicated. Assume that all variables represent positive numbers. 1. 102823212 ++− 2. 5626863 −−+
3. 244520542 −+− 4. xxx 2169 +−
5. 163
643
+ 6. xxx 11819 22 −+
7. 44 51024040 xxxx −−+ 8. xxxx 23505472 2 −−+
9. 333 259592 −+ 10. 3 33 3 813273 zz +
11. 33 51024040 xxxx −−+ 12. 332 23505472 −−+ xx
107
7.4 Multiplying and Dividing Radicals Learning Objectives:
1. Multiply radicals. 2. Divide radicals. 3. Rationalize denominators. 4. Rationalize using conjugates. 5. Key Vocabulary: product rule for radicals, quotient rule for radicals, rationalizing,
conjugates. A. Multiplying and Dividing Radicals Radical Properties. If a and b are real numbers, 0b ≠ , then 1. abba =⋅ 2. aaaaa =⋅=⋅
3. ba
ba= 4. ( ) cabacba +=+
5. ( ) acabcba +=+
Examples 1. Multiply and simplify. Assume that all variables represent positive real numbers. 1. x15x5 3 ⋅ 2. ( )2x3 3. ( )236 + 4. ( )( )259958 ++
108
5. ( )2834 −
6. 2
150
7. xy
yx2
24 43
B. Rationalize denominators—is to rewrite the expression so that the denominator does not
contain a radical expression. Step to rationalize denominators
1. Simplify the radical if needed. 2. Multiply the numerator and the denominator by the radical in the denominator. 3. If there is two terms in the denominator, multiply the it’s conjugate. a. ba + its conjugate ba − b. ba − its conjugate ba +
Example 2. Rationalize each denominator and simplify. Assume that all variables represent
positive real numbers.
1. 5
7
109
2. 125
3. 36
2
−
7.4 Exercise Multiply or divide and simplify completely. Assume that all variables represent positive real numbers. 1. 142 ⋅ 2. ( )28 y 3. yxyx 222 218 ⋅
4. ( )7310 − 5. ( )( )5257 −+ 6. ( )( )13235 −+
7. ( )223 −y 8. 3
120 9. yx
yx2
35
3
96
Rationalize each denominator and simplify. Assume that all variables represent positive real numbers.
10. 32 11.
185 12.
yy
35
13. 12
3+
14. x+3
9 15. 23
13−+
110
7.5 Solving Radical Equations Learning Objectives:
1. Solve radical equations by using the squaring property of equality once. 2. Solve radical equations by using the squaring property of equality twice. 3. Key Vocabulary: radical equations.
Properties for solving radical equations 1. If ax = then 22 ax =
2. If ax = then 2ax =
Steps for solving radical Equations 1. Isolate the square root terms containing the variable. 2. Square both sides of the equation. 3. Simplify both sides of the equation 4. Solve the resulting equation. 5. Check all possible solutions in the original equation. Example 1. Solve: 1. 51 =+x 2. 052 =+−y
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7.5 Exercise Solve each equation. 1. 25 =+x 2. 253 =+x 3. 583 =+x
4. 845 +=− xx 5. xxx 42216 2 =++ 6. xxx 2234 2 =++
7. 174 +−= xx 8. 356 =++x 9. 624 += xx
10. 122 +−= xx 11. 481 =−− xx 12. xx =−+ 152
13. 39 −=+ xx 14. 28 −=− xx 15. 4333 +=+− xx
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7.6 Radical Equations and Problem Solving Learning Objectives:
1. Use the Pythagorean Theorem to solve problems. 2. Solve problems using formulas containing radicals. 3. Key Vocabulary: right triangle, Pythagorean Theorem.
A. Using the Pythagorean Theorem
Pythagorean Theorem—In any right triangle
222 cba =+ Example 1. Use the Pythagorean Theorem to find the length of the unknown side of each right
triangle. Give an exact answer. 1.
Example 2. Find the length of the unknown side of each right triangle with sides a, b, and c,
where c is the hypotenuse. Give an exact answer. 1. a = 15, b = 20 Example 3. Solve. Give an exact answer. 1. A baseball diamond is a square measuring 90 feet on each side. What is the distance from
first base to third base?
a
b
c
6
14
c
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2. A firefighter places a 37‐foot ladder 9 feet from the base of the wall of a building. Will the top reach a window ledge that is 35 feet above the ground?
B. Solving Applications Containing Radicals Formula Example 4. The formula ghv 2= is used to determine the velocity, v, in feet per second, of an
object after it has fallen at a certain height, g is the acceleration due to gravity and h is the height the falling object. On Earth, the acceleration g due to gravity is approximately 32 feet per second. Find the velocity of an object after it has fallen 20 feet.
7.6 Exercise Use the Pythagorean Theorem to find the unknown side of each triangle. Give an exact answer. 1. 2. 3. Find the length of the unknown side of each right triangle with sides a, b and c where c is the hypotenuse. Give an exact answer. 3. 6,3 == ba 4. 10,20 == ca 5. 76,22 == cb
6. A 26‐inch diagonal TV screen is 20 inches wide. Find its height. Give an exact answer.
7. A 18‐inch diagonal computer monitor screen is 12 inches high. Find its width. Give an exact answer. Police use the formula fds 30= to estimate the speed s of a car in miles per hour, given the distance d the car skidded and the type of road surface .f For wet concrete, f is 0.35.
5 5
14 8
9
6
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8. Find how fast a car is going if it skidded 193 feet on wet concrete. Give an exact answer.
9. Find how far a car will skid if the driver tries to stop when going 65 miles per hour on wet cement. Give an exact answer.
The formula 4dt = related the distance d, in feet, an object falls in t seconds, assuming air
resistance does not slow the object down. 10. Find how far, to the nearest tenth of a foot, an object falls if it takes 7.2 seconds to hit the ground. Give an exact answer. The maximum distance d, in kilometers, that you can see from a height of h meters is hd 5.3= 19. Suppose you are riding a glass elevator up the side of the building. Find your height when you first see a landmark you know is 17 kilometers away. Give an exact answer. 20. From the top of a building you can see 50 kilometers. Find the height of the building. Give an exact answer.
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Chapter 8 Quadratic Equations 8.1 Solving Quadratic Equations Using the Square Root Property Learning Objectives:
1. Review factoring to solve quadratic equations. 2. Use the square root property to solve quadratic equations. 3. Use the square root property to solve applications. 4. Key Vocabulary: zero factor property, square root property
A. Solving Quadratic Equation using the Square Root Property Quadratic Equations—is an equation of the form: 02 =++ cbxax Square Root Property of Equations: If a is a positive number and ax =2 then ax ±= Example 1. Solve the following. 1. 132 =x 2. 0349 2 =−x 3. ( ) 366 2 =+x 4. ( ) 05316 2 =+−x
117
5. 0722316
2
=−⎟⎠⎞
⎜⎝⎛ −y
B. Applications of Quadratic Equations Example 2. Solve the following applications. Give exact answer. 1. Use the formula 216h t= to solve the following: determine the time of a stuntman’s fall if he jumped
from a height of 450 feet.
2. Use the formula for the area of a square 2A s= where s is the length of a side. If the area of a
square is 40 square inches, find the length of the side.
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8.1 Exercise Use the square root property to solve each quadratic equation.
1. 252 =x 2. 6412 =x 3. 492 −=x
4. 310 2 =x 5 0112 =−x 6. 471 2 =x
11. ( ) 312 2 =−x 12. 161
41 2
=⎟⎠⎞
⎜⎝⎛ −m 13. ( ) 25112 2 =−y
14. ( ) 8074 2 =−x
15. The area of a circle is found by the equation 2rA π= . If the area of a certain circle is 20π square meters, find its radius. Give exact answer. 20. A square gymnastics mat has a diagonal of 35 feet. Find the length of each side. Give exact answer.
x 35 ft.
x
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8.2 Solving Quadratic Equations by Completing the Square Learning Objectives:
1. Solve quadratic equations of the form 2 0x bx c+ + = by completing the square. 2. Solve quadratic equations of the form 2 0ax bx c+ + = by completing the square. 3. Key Vocabulary: completing the square, coefficient, perfect square trinomial
A. Completing the Perfect Square Recall a perfect square: A perfect Square: Steps to make the expression a perfect square • Divide the coefficient of x term by 2, and then square the result to obtain the last term. Example 1. Find the missing term(s) to make the expression a perfect square. 1. +− xx 122 = ( )2 2. ++ xx 52 = ( )2 B. Solving Quadratic Equation by Completing the Square: 02 =++ cbxax Steps for solving quadratic by completing the square:
1. Rewrite the equation to the form cbxax 2 −=+ . 2. The coefficient of 2x must be 1.
3. Compute 2
2⎟⎠⎞
⎜⎝⎛ b and add to both sides of the equation.
4. Rewrite the left hand‐side as a perfect square. 5. Use the square root property ( )axax ±=⇒=2 to solve the resulting equation.
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Example 2. Solve by completing the square. 1. 027244 2 =+− xx 2. 0463 2 =++ xx 3. ( )( ) 614 −=−+ xx
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8.2 Exercise Solve each quadratic equation by completing the square. 1. 27122 −=+ xx 2. 492 =− xx 3. 0742 =−+ xx
4. ( ) 307 =−xx 5. 36123 2 =− xx 6. 5124 2 += xx
7. 81742 2 =++ xx 8. 0152 2 =+− xx 9. 03147 2 =++ xx
10. 0102 =− xx
122
8.3 Solving Quadratic Equations by Quadratic Formula Learning Objectives:
1. Identify the values of a, b, c in the quadratic equations. 2. Use the quadratic formula to solve quadratic equations. 3. Approximate solutions to quadratic equations. 4. Key Vocabulary: quadratic formula, standard form.
Quadratic Formula: The solutions of 02 =++ cbxax , 0≠a are Steps for solving quadratic equation using quadratic formula:
1. Rewrite the quadratic equation in standard form 02 =++ cbxax . 2. If the equation contains fraction, then clear the fraction first. 3. Identify a, b and c. 4. Substituting these values in the quadratic formula.
Example 1. Solve. 1. 442 += xx
2. 41
83
4
2
=− xx
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8.3 Exercise Use the quadratic formula to solve each quadratic equation. 1. 0762 =−− xx 2. 03102 2 =++ xx 3. 0136 2 =−x
4. xx =− 562 5. 147 2 =x 6. 223 2 −= xx
7. xx −= 14 2 8. 72
715 2 =− xx 9.
4134 2 =− xx
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8.4 Graphing Quadratic Equations Learning Objectives:
1. Graph quadratic equations of the form 2y ax= . 2. Graph quadratic equations of the form 2y ax bx c= + + . 3. Key Vocabulary: parabola, vertex, symmetric about the yaxis, line of symmetry, xintercept,
yintercept, vertex formula.
The graph of cbxaxy ++= 2 , 0≠a is parabola that 1. Open upward if a is positive. 2. Open downward if a is negative.
A. Graphing Parabolas by Plotting Points Example 1. Graph and label at least 5 points on the graph grid.
1. 2
21 xy −=
-10
y
–10
10
10
x
x
y
y-intercept
x-intercept
Line of symmetry
Vertex x
y
y-intercept x-intercept
Line of symmetry
Vertex
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2. 22 −= xy B. Graphing Parabolas Using the Intercepts and the Vertex Vertex—is the lowest point on a parabola that opens upward or the highest point on a parabola that opens downward. Steps for graphing a factorable quadratic equations. 1. Find the y‐intercept by letting 0=x and then solve for y. 2. Find the x‐intercept by letting 0=y and then solve for x.
3. Find the vertex by findabx
2−
= , then substitute the result into the equation and solve for y.
4. Plot the points found in steps 1—3 and draw a parabola. Example 1. Graph the following equation. Label the x, y‐intercepts and the vertex. 1. 962 +−= xxy
-10
y
–10
10
10
x
-10
y
–10
10
10
x
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2. 322 +−−= xxy 8.4 Exercise Graph each quadratic equation by finding and plotting five ordered pair solutions.
1. 24xy = 2. 22xy −= 3. 2
32 xy =
For each equation by a) Find the y‐intercept b) Find the x‐intercept(s) c) Find the vertex d) Sketch the graph and label all points found in (a)‐(c). 4. 782 ++= xxy 5. 322 −−= xxy 6. 862 −+−= xxy
7. 442 −+−= xxy 8. 432 −−= xxy 9. 652 ++= xxy
10. 542 −−= xxy
-10
y
–10
10
10
x