ReteachAdd Integers

To add integers with the same sign, add their absolute values. The sum is:

• positive if both integers are positive.

• negative if both integers are negative.

To add integers with different signs, subtract their absolute values. The sum is:

• positive if the positive integer's absolute value is greater.

• negative if the negative integer's absolute value is greater.

To add integers, it is helpful to use a number line.

CD't1D Find 4 + (-6). ~ Find -2 + (-3).

Use a number line.

• Start at O.• Move 4 units right.• Then move 6 units left.

I II -6 II " I

: +4:I • i• I I I I I .

-3 -2 -1 0 1 2 3 44 + (-6) =-2

1 •

5

Use a number line.

• Start at O.• Move 2 units left.• Move another 3 units left.

i -3 i:' 1~2I I

'1' II ~ II·-6-5-4-3-2-1 0 1 2 3

-2+ (-3)=-5

Add.

1. -5 + (-2)

Write an addition expression to describe each situation. Then findeach sum.

13. HAWKA hawk is in a tree 100 feet above the ground. It flies down to theground.

ReteachSubtract Integers

6 - 9 = 6 + (-9) To subtract 9, add -9.= -3 Simplify .

••• Find -10 - (-12).

-10 - (-12) = -10 + 12 To subtract -12, add 12.= 2 Simplify.

~ Evaluate a - b i~a = -3 and b = 7.

a - b = -3 - 7= -3 + (-7)= -10

Replace a with -3 and b with 7.

To subtract 7, add -7.Simplify.

Subtract.

1. 7 - 9

ReteachMultiply Integers

The product of two integers with different signs is negative.

The product of two integers with the same sign is positive.

~ Find 5(-2).

~ Find -3(7).

~ Find -6(-9).

~ Find (-7)2.

(-7)2 = (-7)(-7)= 49

There are 2 factors of -7.The product is positive.

~. Evaluate abc if a = -2, b = -3, and c = 4.

abc = -2(-3)(4)= 6(4)= 24

Replace a with -2, b with -3, and c with 4.MUltiply -2 and -3.Multiply 6 and 4.

crmD.. ... - .........•...........

ReteachDivide Integers

The quotient of two integers with different signs is negative.

The quotient of two integers with the same sign is positive.

~ Find30+(-5).

30 + (-5)

30 + (-5) = -6

The integers have different signs.

The quotient is negative.

~ Find-100 + (-5).

-100 + (-5) The integers have the same si~n.

-100 + (-5) = 20 The quotient is positive.

3 18. -2

6 -80• -20

9 540• 45 10 -256

• 16

11. 12 + e 12. 40 + f

13. d + 6 14. d + e

15. f + e 16. e2 + f

17. ~d 18. ef + 2

19. f~48 20. d ~ e

ReteachAdd and Subtract Unlike Fractions

To add or subtract fractions with different denominators,

• Rename the fractions using the least common denominator (LCD).

• Add or subtract as with like fractions.

• If necessary, simplify the sum or difference.

~Find:+~.

Method 1 Use a Model.23

+ 14

1112

Method 2 Use the LCD.2 124 1 3-+-=-.-+-.-3 4 3 4 4 3

8 3 11= 1:2+ 1:2or 1:2

Add or subtract. Write in simplest form.1 3 3 11. 2+4 2.8-2

3. 175+ (- ~) 2 14. "5-3

5. ~ + (- :2)11 3

6·1:2-4

7. ~ - (- ~)7 18.9-2

3 7 3 29.10+1:2 10."5+ 3

III

- - - - - - T - - - - - - - r - - -- - --.I I II I I_______ L L I

ReteachMultiply Fractions

To multiply fractions, multiply the numerators and multiply the denominators ..§..x~=5x3=~=-.16 5 6x5 30 2

To multiply mixed numbers, rename each mixed number as an improper fraction. Then multiply thefractions.

~ Find ~ X :. Write in simplest form.

~ x .!= 2 x 4 +- Multiply the numerators.3 5 3 x 5 +- Multiply the denominators.

~ Find ~ X ~. Write in simplest form.

1.. x 21.. = 1.. x -2.. Rename 2-2

1as an improper fraction, ~2'3 2 3 2

1 x 53x256

Multiply. Write in simplest form.2 2 1 71. g xg 2.2" xs

54."9 x 4 5. 1~ x (- ~)

1 411. -23 x"6 1 312.8 x 24

Reteach

To divide by a fraction, multiply by its multiplicative inverse or reciprocal. To divide by a mixed number,rename the mixed number as an improper fraction .

.~. Find ~ + :. Write in simplest form.

st + ~ = ~o+ ~ Rename 3t as an improper fraction,

10 9 2 9=3 .2" MUltiply by the reciprocal of 9' which is 2'

5 }=1-.% Divide out common factors.

1 1

= 15 MUltiply.

Divide. Write in simplest form.

12.1 22.5'374 '576 3 _l ~l

• 2' 5

4. 5 + (- ~) 5. ~ + 10 6. 7~ + 2C"l0'tl

~.'g:@

9~0

7 5 . 31 8. 36 + 1~ 9. -2~ + (-10)~

• 67 2" ~~~"ee$.l!J.0:;

10. ~ + 1~ 11. 6~ +* 12·4+ :g,

~'"a::"~~~C"l

13. 4~ + 2~ 14. 12 + (-2~) 15. 4+ ~0S'tl

~.!"5"'"

Study Guide and InterventionFractions and Decimals

To write a decimal as a fraction, divide the numerator of the fraction by the denominator. Use a power often to change a decimal to a fraction.

Write : as a decimal.

Method 1 Use pencil and paper.

0.555 ...9)5.000

4550455045

5

You can use bar notation 0.5 toindicate that 5 repeats forever.

5 -So, "9 = 0.5.

Write 0.32 as a fraction in simplest form.

320.32 = 100

_ 8- 25

Write each fraction or mixed number as a decimal. Use bar notationif the decimal is a repeating decimal.

476. 399

Practice: Skills

Write each fraction or mixed number as a decimal. Use bar notationif the decimal is a repeating decimal.

1710. 437

713. 632

195. 20

58. 18 199.2"5

1712. 24

1715. 148

Study Guide and InterventionFractions and Percents

Write ~~as a percent.

A ratio is a comparison of two numbers by division. When a ratio compares a number to 100, itcan be written as a percent. To write a ratio or fraction as a percent, find an equivalent fraction witha denominator of 100. You can also use the meaning of percent to change percents to fractions.

/X5",19 95 _ 95nt Since 100 -7- 20 = 5, mUltiply the numerator andl'l7\ 'l7\7\ - 70£.V .LVV denominator by 5.

'" X5/

9292% = 100

_ 23- 25

141. 100

272. 100

37. 100

148. 100

110. 20

1311. 25 4

12. 10

Study Guide and InterventionPercents and Decimals

To write a percent as a decimal, divide the percent by 100 and remove the percent symbol. To write adecimal as a percent, multiply the decimal by 100 and add the percent symbol.

42.5% = i~·g42.5 x 10100 x 10425

= 1,000= 0.425

0.625 = 0§65= 62.5%

Multiply by 100.

Add the % symbol.

Write each percent as a decimal.

1. 6%

Fractions, Decimals, and Percents

1. 12% 2. 125% 3. 96%

4. 1.7% 5. 5.75% 6. 45%

7. 0.36 8. 1.94 9. 0.0425

10. 0.5 11. 0.85 12. 5.2

64 5 1813. 100 14. "8 15. 20

41 3 716. 50 17. '7 18. 10

ReteachSolve Two-Step Equations

To solve a two-step equation, undo the addition or subtraction first. Then undo the multiplication ordivision.

7v - 3 = 25+3 = +3

7v = 287v 28=7 7v=4

Write the equation.Undo the subtraction by adding 3 to each side.Simplify.

7v - 3 = 257(4) - 3 ~ 2528 - 3 ~ 2525 = 25 ..(

Write the original equation.Replace v with 4.Multiply.The solution checks.

The solution is 4.

Solve -10 = 8 + 3x. Check your solution.

-10= 8+ 3x-8= -8

-18= 3x-18 3x3 3-6=x

Simplify.

Undo the multiplication by dividing each side by 3.

Simplify.

Write the equation.Undo the addition by subtracting 8 from each side.

-10= 8 + 3x-10~8+3(-6)-10~ 8 + (-18)-10 = -10 ..(

Write the original equation.Replace x with -6.Multiply.The solution checks.

1. 4y + 1= 13 2. 6x + 2 = 26 3. -3 = 5k + 7 24. 3"n + 4 = -26

5.7= -3c - 2 6. -8p + 3 = -29 7. -5 = -5t - 5 8. -9r + 12= -24

7 10. 35 = 7 + 4b 4 12. 49= 16+ 3y9.11 + gn = 4 11. -15 + -p = 95

13.2 = 4t - 14 14. -9x - 10= 62 15. 30= 12z - 18

ReteachSolve Equations with Variables on Each Side

To solve an equation with variables on each side, use the Properties of Equality to write an equivalentequation with the variables on one side.

Then solve the equation.

6x + 10 - 10 = x + 8 - 10

6x=x-2

4x = x + 274x - x = x - x + 27

3x = 273x 27- -3 3x=9

Solve 3x - 16 = 5x - 4.

3x -16 = 5x - 4

3x - 3x - 16 = 5x - 3x - 4

-16 = 2x - 4

-16 + 4 = 2x - 4 + 4

-12 = 2x12 2x-- -2 2-6 =x

Write the equation.

Subtraction Property of Equality

Simplify.

Division Property of Equality

Simplify. Check your solution.

Subtraction Property of Equality

Simplify.

Addition Property of Equality

Simplify.

Division Property of Equality

Simplify. Check your solution.

Express each equation as another equivalent equation. Justify youranswer.

Solve each equation. Check your solution.

3. 2x + 6 = x + 15

5. -12 - x = 8 - 3xChapter 3

Skills PracticeSolve Equations with Variables on Each Side

Express each equation as another equivalent equation. Justify youranswer.

6. ~ - 12 = -3 + x2

11. ~- 3 = x + 84