algebra notes

Algebra Notes

Writing Algebraic Expressions

Let Statement: math sentence used to define a variable to represent the unknown quantities.

Laura has twice as much homework as Ann.

The Bills won five more games than they lost.

Seven more than three times a number is 25.

The length of a rectangle is 3 cm more than the width.

Let Ann = a Let Laura = 2aLet games lost = g Let Bills won = 5 + gLet Yankees = y Let Tigers = 3yLet width = wLet length = 3 + w

Mike is three years older than Jim.

Eight more than twice a number is 32

Seven more than three times a number is 25.

Twice a number increased by four is 16.

Let Jim = j Let Mike = 3+ jLet number = n 2n + 8 = 32Let number = n 3n + 7Let number = n 2n + 4 = 16

Six less than three times a number is 21.

Fifteen less than twice a number is 25.

Sixty-six is eleven more than five times a number.

Let number = n3n 6 = 21Let number = n2n 15 = 25Let number = n66 = 5n + 66

Writing Algebraic Expressions

Setting up & Solving word problems

Write your let statementWrite your equationSolve CheckWrite an answer sentence

A cell phone company charges $39 a month plus $.15 per text message sent. If Jan sends 35 text messages this month, how much does she owe before taxes are added?

The Bills won five more games than they lost.

Let text message = t 39 + 0.15tt = 44.25 Jam owes $44.25 Let text message = s 12 + 2ss = 4 4 snacks

A rental car company ABC charges $25 per day plus $.15 per mile. Rental car company XYZ charges $18 per day plus $.25 per mile. If you plan to drive 50 miles, who is the cheaper rental company?

Joe attends a carnival. The admission is $48. Tickets for rides cost $4 each. Joe needs one ticket for each ride. Write an equation Joe can use to determine the number of ride tickets, r, he can buy if he has $200 before he pays the admission fee.

Let miles = m ABC: 25 + 0.15m$32.50 XYZ is cheaper Let number or rides = r 48 + 4r = 200r = 38 38 rides XYZ: 18 + 0.25$30.50

substitutions

Evaluate if s = 44s

4 + s

5 s

12 s

4(4)16 4 + 48 5 - 4112 43

Evaluate if s = -67s

3 + s

7 s

18 s

7(-6)-42 3 + (-6)-3 7 (-6)1318 (-6)-3

Evaluate if n = 3 and r = 5n + 7r

9n - r

2nr + 6n

3 + 7(5)9 +1221 9(3) - 527 2522(3)(5) + 6(3)30 + 1848

Evaluate if p = 12 and q = -8p + q +6

p q + 3

p q + q

12 + (-8) + 6-4 + 6 2 12 (-8) + 3 20 + 32312 (-8) + (-8)20 + 6484

Evaluate if a = -2 and b = 63a + 5b

4a + 3b

7a - (b/3)

3(-2) + 5(6)3(4) + 5(36)1924(-2) + 3(6)4(-8) + 18-147(-2) - (6/3)7(4) (36/3)28 1216

Like Terms

Terms of an ExpressionTerms are parts of a math expression separated by addition or subtraction signs.

3x + 5y 8 has 3 terms.

Like TermsLike Terms: have the same variables to the same powers 8x+2x+5a +a 8xand 2x are like terms5a and a are like terms

LIKE terms: Yes or No?3x + 7xYes - Like5x + 5yNo - Unlike4c + cNo - Unlike4d + 4Yes - Like

LIKE terms: Yes or No?3ab 6bYes - Like2a 5aNo - Unlikex and xNo - Unlike6 and 10Yes - Like

Identify the LIKE terms3m 2m + 8 3m + 65x + b 3x + 4 + 2x 1 3b-6y + 4yz + 6x + 2yz 4y + 2x - 5

CoefficientsA Coefficient: a number written in front of the variable.Example: 6xThe coefficient is 6.Example: xThe coefficient is 1.

Combine LIKE terms by adding their coefficients.Simplify: means to combine like terms.Simplify

+3c + 4c=7cWrite an expression:

-8a - 1a= 7aWrite an expression:

+5c + 4dWrite an expression:

-5a 4bThis expression cannot be simplified. Why not?Write an expression:

Simplify the following expressions

2x + 4x

2a + 5a + 6

3xy xy +2x

-4c + 8c 6c

3a + 7a

3y + 5y -4y

cd + 4cd 2a

e 2e + e

6xy 2xy

5d 6d 3d

4s 4s

5x + 4x + 4x + 11x

6x 10a 4xy 7a + 6 4y -4d 2xy + 2x 5cd 2a 0 24x -e -2c

Challenge questions

1.5x 3x2.8x 2x3.7x (3x)4.6x (4x)5.10x 14x6.9x (x)

7.3x 8x8.x (5x)

9. a + b + 2a + 5b10. 7h + 3 2h + 4

-8x6x-4x10x-24x-8x-5x6x3a + 6b5h + 7

11. 3x + 3y + x + y + z 12. 5b +5b + 6b - 10 3b

13. Find the perimeter of the rectangle:A 4x + 3yB 8x + 6yC 12xyD 4x+ 3y

4x + 4y +z6a + 7b - 10

Adding & subtracting Polynomials

AddingCombine like termAdd the coefficients to simplify

Example: Add 2x + 6x + 5 and 3x - 2x 1

Start with:2x + 6x + 5 + 3x - 2x 1

Place like terms together:_______+ ________+ ________

Add the like terms: _________+ __________+ _________

Final answer:4x6x 2x5 1 5x2x - 3x45x + 4x + 4

SubtractingChange the subtraction sign to addition and reverse the sign of each term that followsThen add as usual

Example: Subtract 5y + 2xy - 5 and 3x - 2x 1

Start with:5y + 2xy - 5 - 2y - 3xy + 3

Place like terms together:_______+ ________+ ________

Add the like terms: _________+ __________+ _________

Final answer:-xy2xy 3xy-5 + 33y5y - 2y-23y - xy - 2++- -

Try the following

1. (2x + 3y) + (4x + 9y) 2.(3a + 5b + 7c) - (5a 2b + 9c)

3. (3x 5) + (x 7) + (7x + 12)

4.(3a + 5b + 7c) + (8a 2b 9c)

5. 4x + 6x 8x 10 and 7x 4x + 9x + 3

6. Subtract (5m 6n + 12) from (2m + 3n 5).

(2m + 3n 5) - (5m 6n + 12) 6x + 12y-2a + 7b 2c11x11a + 3b 2c 3x + 2x - x - 7-3m + 9n -17

7.Subtract 8a + 5b 6c from 10a + 8b + 7c(10a + 8b + 7c) - (8a + 5b 6c)

8. (4x + 8y + 9z 7a + 5b) (4b + 5x + 7y + 3z + 2a)

9. ( 3x2 + 4x 11) (6x2 8x + 10).

10. (7e + 3e +2) + (9 6e + 4e) + (9e + 2 6e)

2a 3b + 13c-x y + 6z 9a + b3x + 12x - 215e + 6e + 13

challenge

Some of the measures of the polygons are given. P represents the measure of the perimeter. Find the measure of the other side or sides.2x + y14x - 4x + 7x - 15x + 34x - 3

The distributive property

The Distributive PropertyDistributive Property: the process of distributing the number on the outside of the parentheses to each term in the inside.a(b + c) = ab + ac Example:

5(x + 7) =5x + 355x57+

Practice #13(m - 4)3 m - 3 43m 12

Practice #2-2(y + 3)-2 y + (-2) 3-2y + (-6)-2y - 6

3(x + 6) =3x + 184(4 y) =16 4y7(2 + z) =14 + 7z5(2a + 3) = 10a + 15Simplify the following:

6(3y - 5) =18y 30 3 +4(x + 6) =4x + 272x + 3(5x - 3) + 5 = 17x 4 Simplify the following:

Distributive practice

2(4 + 9x) 2. 7(x + -1) 3. 12(a + b + c)

7(a + c + b) 5. -10(3 + 2 + 7x) 6. -1(3w + 3x + -2z)

-1(x + 2) 8. 3(-2 + 2x2y3 + 3y2) 9. 5(5 + 5x)

y(1 + x) 11. 12x(3x + 3) 12. 9(9x + 9y)

8 + 18x7x - 712a + 12b + 12c7a + 7c + 7b-70x - 50-3w 3x + 2z-x 2 -6 + 6xy + 9y25x + 25y + yx36x + 36x81x + 8y

Factoring

factoringTo factor expressions find the GCF (greatest common factor) of the terms Factoring is the opposite of distributing.

Find the GCF of each pair of monomials 4x, 12x 2. 18a, 20ab 3. 12cd, 36cd

4x2a12cd

Factor each expression4. 12a 6h 5. 3x + 9 6. 12x + y

7. 24a 4 8. 72a + 9n 9. 8a - 8v

6(2a h)3(x + 3)Cannot be simplified4(6a 1)9(8a + n)8(a v)

Solving equations

Steps to Solving Equations

2n 10 = 50 Equation: a mathematical sentence that uses an equal (=) sign.Step 1: Get rid of the 10. Look at the sign in front of the 10, since it is subtraction we need to use the opposite operation (addition) to cancel out the 10Add 10 to both sides. Remember, what you do to one side of the equation, you have to do to the other.

+10+102n = 60


2n = 60Step 2: Next, we need to look at what else is happening to the variable. 2n means that two is being multiplied to n, therefore we need to do the opposite (division) to undo the multiplication.Divide both sides by 2. Remember, what you do to one side of the equation, you have to do to the other.

22n= 30


2n 10 = 50Step 3: CHECK your solution!! First, rewrite the original equationWe already solved for n, so wherever you see the variable, n, plug in the answer.Evaluate the equation, SHOWING ALL WORK!Does it check?2 (30) 10 = 5060 10 = 5050 = 50

Solve & Check

105 = 10n + 5

n/5 + 3 = 6

-44 + 7n = 250

-1/2 = -5/18h

200 = 100 25n

-9.4 + z = -3.6n = 42n = 10n = -4n = 15h = -9/5z = 5.8

Solving equations practice

x 3 = 192.a 14 = 6

3. 9x = 634.5x 2 = 8

6. 8a + 5 = 53

-7 = c 68.a 3.5 = 4.9

9. x 2.8 = 9.510. 2.25 + b = 1.

14.2(b 2) + b + 3

x = 22a = 22x = -30x = 22a = 22x = 22a = 22b = 22x = 12.3c = 22

11.12. -8.5 + r = -2.1

13. 14. 2(b 2) + b = 6.5

.

c = 1 3/7m = 33/14r = 6.4b = 2.5

Solving multi-step equations

Steps to Solving Multi-Step Equations

4(n 5) - 7 = 9 + 2n 4nStep 1: Distribute if necessary variable. Distribute the 4 to the n and 5.

4n 20 - 7 = 9 + 2n 4n


Step 2: Combine like terms on each side of the equations.On the left side -20 and -7 combine to get -27On the right side 2n and -4n combine to get -2n

4n 20 - 7 = 9 + 2n 4n4n 27 = 9 2n


Step 3: Get all variables to one side of the equation.First we want to get rid of the -27. Look at the sign in front of -27, since it is subtraction (or a negative) we need to use the opposite operation (addition) to cancel it out. Therefore add 27 to both sides.4n = 36 2n4n 27 = 9 2n+27+27

Steps to Solving multi-step Equations

Step 4: Get all plain numbers to one side of the equationFirst we want to get rid of the -2n. Look at the sign in front of -2n, since it is subtraction (or a negative) we need to use the opposite operation (addition) to cancel it out. Therefore add 2n to both sides.4n = 36 2n +2n+2n6n = 36


Step 5: Next, since we have all the variables on one side and all the plain numbers on the other side we need to look at what else is happening to the variable.6n means the 6 is being multiplied by n, therefore we need to do the opposite (division) to undo the multiplication. So, divide both sides by 6.n = 6666n = 36


Step 6: CHECK your solution!! First, rewrite the original equationWe already solved for n, so wherever you see the variable, n, plug in the answer.Evaluate the equation, SHOWING ALL WORK!Does it check?4 7 = 21 - 244(n 5) - 7 = 9 + 2n 4n4(6 5) - 7 = 9 + 2(6) 4(6)4(1) - 7 = 9 + 12 24-3 = -3

Solve & Check

9 + 5r = -17 8r

3(n + 5) + 2 = 26

58 + 3y = -4y 19

4 2(v 6) = -8

r = -2v = 12n = 3y = -11

Inequalities

Inequality: a mathematical sentence using , , or .Example: 3 + y > 8.

Inequalities use symbols like < and > which means less than or greater than.

They also use the symbols and which means less than or equal to and greater than or equal to.Inequalities

Whats the difference?x < 4 means that x is less than 4 4 is not part of the solutionWhat number is in this solution set?

x 4 means that x can be less than OR equal to 44 IS part of the answerWhat number is in this solution set?

You graph your inequalities on a number line:This graph shows the inequality x < 4

The open circle on 4 means thats where the graph starts, but 4 is NOT part of the graph.

The shaded line and arrow represent all the numbers less than 4.

What is this inequality?X > -2

What is this inequality?X 2 1/2

Use an open circle ( ) to graph inequalities with < or > signs.

Use a closed circle ( ) to graph inequalities with or signs. Graphing inequality solution sets on a number line:

What do you think this symbol means?Does not equal

Example: x 7 means:7 is not equal to x

Graph x -1X -1 would include everything on the number line EXCEPT -1.

Use an open circle to show that -1 is NOT a part of the graph.

Graph x < 4 (a number less than 4)

Graph x < 6 (a number less or equal to 6)

Graph x > 3 (a number less or equal to 6)

Graph each inequality

Graph

x < 3

x > -5

x < -1

x > 2

Solve, Graph, and check each inequality

Solve, Graph, & Check

x + 8 > 15

3y 4 < 11

2x < 18

x + 4 > 2

2n + 7 > 13

n > 3x < 9x > 7y < 5x > -2

Solve, Graph, & Check

5n + 4 < 4n

3x 3 9

n < -4x < -6y 3x 4

*************************NO NOTES!*NO NOTES************************************************

algebra notes

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