Add 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 I I -6 I I " : +4: I I I I I. i -3-2 -1 0 1 2 3 4 4 + (-6) =-2 I 1 5 Use a number line. Start at O. Move 2 units left. Move another 3 units left. i -3 i :' 1~2 I 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 find each sum. 13. HAWKA hawk is in a tree 100 feet above the ground. It flies down to the ground.
Subtract 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
Multiply 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..... -......
Divide 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. e 2 + f 17. ~d 18. ef + 2 19. f ~48 20. d ~ e
Add and Subtract Unlike Fractions To add or subtract fractions with different denominators, Rename the fractions using the least common denominator Add or subtract as with like fractions. If necessary, simplify the sum or difference. (LCD). ~Find:+~. Method 1 2 3 + 1 4 11 12 Use a Model. I I I - - - - - - T - - - - - - - r - - -- - --. I I I I I L L I 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 1 1. 2+4 2. 8-2 3. 1 7 5+ (- ~) 2 1 4. "5-3 5. ~ + (- :2) 11 3 6 1:2-4 7. ~ - (- ~) 7 1 8. 9-2 3 7 3 2 9.10+1:2 10."5+ 3
Multiply Fractions To multiply fractions, multiply the numerators and multiply the denominators....x~=5x3=~=-.1 6 5 6x5 30 2 To multiply mixed numbers, rename each mixed number as an improper fraction. Then multiply the fractions. ~ 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 5 3x2 5 6 Multiply. Write in simplest form. 2 2 1 7 1. g xg 2.2" x s 5 4."9 x 4 5. 1~ x (- ~) 1 4 11. -2 3 x"6 1 3 12. 8 x 2 4
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 '3 7 4 '5 7 6 3 _l ~l 2' 5 4. 5 + (- ~) 5. ~ + 10 6. 7~ + 2 C"l 0 'tl ~. 'g: @ 9 7 5. 31 8. 36 + 1~ 9. -2~ + (-10) ~0 ~ 6 7 2" ~ ~ ~ "ee $. l!j. 0 :; 12 4+ : 10. ~ + 1~ 11. 6~ +* 13. 4~ + 2~ 14. 12 + (-2~) 15. 4+ ~ g, ~ '"a:: "~ ~ ~ C"l 0 S 'tl ~.!" 5"' "
Study Guide and Intervention Fractions and Decimals To write a decimal as a fraction, divide the numerator of the fraction by the denominator. Use a power of ten to change a decimal to a fraction. Write : as a decimal. Method 1 Use pencil and paper. 0.555... 9)5.000 45 50 45 50 45 5 You can use bar notation 0.5 to indicate that 5 repeats forever. 5 - So, "9 = 0.5. 32 0.32 = 100 _ 8-25 Write 0.32 as a fraction in simplest form. Write each fraction or mixed number as a decimal. Use bar notation if the decimal is a repeating decimal. 47 6. 3 99
Practice: Skills Write each fraction or mixed number as a decimal. Use bar notation if the decimal is a repeating decimal. 19 5. 20 5 8. 1 8 19 9.2"5 17 10. 4 37 17 12. 24 7 13. 6 32 17 15. 1 48
Study Guide and Intervention Fractions and Percents A ratio is a comparison of two numbers by division. When a ratio compares a number to 100, it can be written as a percent. To write a ratio or fraction as a percent, find an equivalent fraction with a denominator of 100. You can also use the meaning of percent to change percents to fractions. /X5", Write ~~ as a percent. 19 95 _ 95 nt Since 100-7- 20 = 5, multiply the numerator and l'l7\ 'l7\7\ - 70.V.LVV denominator by 5. '" X5/ 92 92% = 100 _ 23-25 14 1. 100 27 2. 100 3 7. 100 14 8. 100 1 10. 20 13 11. 25 4 12. 10
Study Guide and Intervention Percents and Decimals To write a percent as a decimal, divide the percent by 100 and remove the percent symbol. To write a decimal as a percent, multiply the decimal by 100 and add the percent symbol. 42.5% = i~ g 42.5 x 10 100 x 10 425 = 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 18 13. 100 14. "8 15. 20 41 3 7 16. 50 17. '7 18. 10
Solve Two-Step Equations To solve a two-step equation, undo the addition or subtraction first. Then undo the multiplication or division. 7v - 3 = 25 +3 = +3 7v = 28 7v 28 7 = 7 v=4 The solution is 4. 7v - 3 = 25 7(4) - 3 ~ 25 28-3 ~ 25 25 = 25..( Write the equation. Undo the subtraction by adding 3 to each side. Simplify. Write the original equation. Replace v with 4. Multiply. The solution checks. Solve -10 = 8 + 3x. Check your solution. -10= 8+ 3x -8= -8-18= 3x -18 3x 3 3-6=x -10= 8 + 3x -10~8+3(-6) -10~ 8 + (-18) -10 = -10..( Write the equation. Undo the addition by subtracting 8 from each side. Simplify. Undo the multiplication by dividing each side by 3. Simplify. Write the original equation. Replace x with -6. Multiply. The solution checks. 1. 4y + 1 = 13 2. 6x + 2 = 26 3. -3 = 5k + 7 2 4. 3"n + 4 = -26 5.7= -3c - 2 6. -8p + 3 = -29 7. -5 = -5t - 5 8. -9r + 12= -24 7 4 9.11 + gn = 4 10. 35 = 7 + 4b 11. -15 + -p = 9 5 12. 49 = 16+ 3y 13.2 = 4t - 14 14. -9x - 10= 62 15. 30 = 12z - 18
Solve Equations with Variables on Each Side To solve an equation with variables on each side, use the Properties of Equality to write an equivalent equation with the variables on one side. Then solve the equation. 6x + 10-10 = x + 8-10 6x=x-2 4x = x + 27 4x - x = x - x + 27 3x = 27 3x 27 - - 3 3 x=9 Write the equation. Subtraction Property of Equality Simplify. Division Property of Equality Simplify. Check your solution. Solve 3x - 16 = 5x - 4. 3x -16 = 5x - 4 3x - 3x - 16 = 5x - 3x - 4-16 = 2x - 4-16 + 4 = 2x - 4 + 4-12 = 2x 12 2x -- - 2 2-6 =x 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 your answer. Solve each equation. Check your solution. 3. 2x + 6 = x + 15 5. -12 - x = 8-3x Chapter 3
Skills Practice Solve Equations with Variables on Each Side Express each equation as another equivalent equation. Justify your answer. 6. ~ - 12 = -3 + x 2 11. ~- 3 = x + 8 4