36 Chapter 1 Solving Equations and Inequalities

1.6

How can you use addition or subtraction to

solve an inequality?

Work with a partner. The National Collegiate Athletic Association (NCAA) uses the following formula to rank the passing effi ciency P of quarterbacks.

P = 8.4Y + 100C + 330T − 200N

——— A

Y = total length of all completed passes (in Yards)

C = Completed passes

T = passes resulting in a Touchdown

N = iNtercepted passes

A = Attempted passes

M = incoMplete passes

Which of the following equations or inequalities are true relationships among the variables? Explain your reasoning.

a. C + N < A b. C + N ≤ A c. T < C d. T ≤ C

e. N < A f. A > T g. A − C ≥ M h. A = C + N + M

ACTIVITY: Quarterback Passing Effi ciency11

Touchdown Completed Not Touchdown

Attempts Intercepted

Incomplete

Work with a partner. Which of the following quarterbacks has a passing effi ciency rating that satisfi es the inequality P > 100? Show your work.

Player Attempts Completions Yards Touchdowns Interceptions

A 149 88 1065 7 9

B 400 205 2000 10 3

C 426 244 3105 30 9

D 188 89 1167 6 15

ACTIVITY: Quarterback Passing Effi ciency22

Solving One-Step Inequalities

STANDARDS OF LEARNING

8.15

ANDARDSTAA

Section 1.6 Solving One-Step Inequalities 37

Work with a partner. Use the passing effi ciency formula to create a passing record that makes the inequality true. Then describe the values of P that make the inequality true.

a. P < 0

Attempts Completions Yards Touchdowns Interceptions

b. P + 100 ≥ 250

Attempts Completions Yards Touchdowns Interceptions

c. 180 < P − 50

Attempts Completions Yards Touchdowns Interceptions

d. P + 30 ≥ 120

Attempts Completions Yards Touchdowns Interceptions

e. P − 250 > −80

Attempts Completions Yards Touchdowns Interceptions

ACTIVITY: Finding Solutions of Inequalities33

Use what you learned about solving inequalities using addition or subtraction to complete Exercises 5 – 7 on page 41.

4. Write a rule that describes how to solve inequalities like those in Activity 3. Then use your rule to solve each of the inequalities in Activity 3.

5. IN YOUR OWN WORDS How can you use addition or subtraction to solve an inequality?

6. How is solving the inequality x + 3 < 4 similar to solving the equation x + 3 = 4? How is it different?

Completions Yards Touchdowns Interceptions

38 Chapter 1 Solving Equations and Inequalities

Lesson1.6Lesson Tutorials

Addition Property of Inequality

Words If you add the same number to each side of an inequality, the inequality remains true.

Algebra If a < b, then a + c < b + c.

Subtraction Property of Inequality

Words If you subtract the same number from each side of an inequality, the inequality remains true.

Algebra If a < b, then a − c < b − c.

These properties are true for <, >, ≤, and ≥ .

Study TipYou can solve inequalities in much the same way you solve equations. Use inverse operations to get the variable by itself.

EXAMPLE Solving Inequalities Using Addition or Subtraction11

a. Solve x − 5 < −3.. Graph the solution.

x − 5 < −3 Write the inequality.

+ 5 + 5 Add 5 to each side.

x < 2 Simplify.

The solution is x < 2.

3112345 0 4 52

x 2

Check: x 0 is a solution. Check: x 3 is not a solution.

b. Solve −3.5 ≤ 4.8 + x..

−3.5 ≤ 4.8 + x Write the inequality.

− 4.8 − 4.8 Subtract 4.8 from each side.

−8.3 ≤ x Simplify.

The solution is x ≥ −8.3.

Solve the inequality. Graph the solution.

1. n + 7 ≥ − 4 2. r − 1.2 > − 0.5 3. 3

— 5

≥ z + 2

— 5

Undo the subtraction.

ReadingThe inequality − 8.3 ≤ x is the same as x ≥ − 8.3.

Undo the addition.

Exercises 8 –16

Section 1.6 Solving One-Step Inequalities 39

EXAMPLE Solving Inequalities Using Multiplication or Division22

a. Solve x

— 10

≤ −2.2. Graph the solution.

x

— 10

≤ −2 Write the inequality.

10 ⋅ x

— 10

≤ 10 ⋅ (− 2) Multiply each side by 10.

x ≤ − 20 Simplify.

The solution is x ≤ − 20.

30101020304050 0 40 5020

x 20

Check: x 30 is a solution. Check: x 0 is not a solution.

b. Solve 2.5x > 11.25.11.25. Graph the solution.

2.5x > 11.25 Write the inequality.

2.5x

— 2.5

> 11.25

— 2.5

Divide each side by 2.5.

x > 4.5 Simplify.

The solution is x > 4.5.

0.50 1.51 2.52 3.53 4.54 5.5 65

x 4.5

Check: x 3 is not a solution. Check: x 5 is a solution.

Solve the inequality. Graph the solution.

4. b

— 8

≥ − 5 5. − 0.4 > g —

15

6. 63 < 9q 7. 1.6u > − 19.2

Exercises 21 –29

Undo the division.

Multiplication and Division Properties of Inequality (Case 1)

Words If you multiply or divide each side of an inequality by the same positive number, the inequality remains true.

Algebra If a < b, then a ⋅ c < b ⋅ c for a positive number c.

If a < b, then a

— c <

b —

c for a positive number c.

Undo the multiplication.

40 Chapter 1 Solving Equations and Inequalities

EXAMPLE Solving Inequalities Using Multiplication or Division33

a. Solve y —

−4 > 6.6. Graph the solution.

y —

− 4 > 6 Write the inequality.

− 4 ⋅ y —

− 4 < − 4 ⋅ 6 Multiply each side by − 4.

Reverse the inequality symbol.

y < − 24 Simplify.

The solution is y < − 24.

4122024283236 16 0 48

y 24

Check: y 28 is a solution. Check: y 0 is not a solution.

b. Solve −21 ≥ −1.4y..

− 21 ≥ − 1.4y Write the inequality.

− 21

— − 1.4

≤ − 1.4y

— − 1.4

Divide each side by − 1.4. Reverse the inequality symbol.

15 ≤ y Simplify.

The solution is y ≥ 15.

Solve the inequality. Graph the solution.

8. 7 > j —

− 1.5 9.

a —

− 3 ≤ − 2

10. − 2s < 24 11. − 3.1z ≥ 62

Exercises 31 –39

Undo the division.

Multiplication and Division Properties of Inequality (Case 2)

Words If you multiply or divide each side of an inequality by the same negative number, the direction of the inequality symbol must be reversed for the inequality to remain true.

Algebra If a < b, then a ⋅ c > b ⋅ c for a negative number c.

If a < b, then a

— c >

b —

c for a negative number c.

Undo the multiplication.

A negative sign in an inequality does not necessarily mean you must reverse the inequality symbol. Only reverse the inequality symbol when you multiply or divide both sides by a negative number.

Common Error

Section 1.6 Solving One-Step Inequalities 41

Exercises1.6

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

1. REASONING How are inequalities and equations the same? different?

2. WRITING Explain how solving 3m < − 9 is different from solving − 3m < 9.

3. OPEN-ENDED Write an inequality that is solved using the Subtraction Property of Inequality.

4. WHICH ONE DOESN’T BELONG? Which inequality does not belong with the other three? Explain your reasoning.

2n ≥ 10

10 ≤ 2n

− 2n ≤ − 10

10 ≥ − 2n

Use the formula in Activity 1 to create a passing record that makes the inequality true.

5. P ≥ 180 6. P + 40 < 110 7. 280 ≤ P − 20

Solve the inequality. Graph the solution.

8. g + 8 > 5 9. m − 6 < 4 10. − 10 ≤ x − 3

11. k − 2.9 ≥ 1.5 12. 3.6 ≤ w + 5.8 13. c − 1

— 4

> − 3

— 4

14. b + 2

— 3

≥ − 1

— 2

15. m − 4.7 < − 12.3 16. 6 ≥ x + 2

— 5

17. ERROR ANALYSIS Describe and correct the error in solving the inequality.

18. WATER A dog’s water container holds at most 20 quarts.

a. Which inequality shows how much water w your dog has drunk? Solve the inequality.

w + 16 ≥ 20 w + 16 ≤ 20

b. Interpret the solution to part (a).

Help with Homework

11

2.5 > x + 6.2 6.2 − − 6.26.2 − − 6.26.2 −3.7 3.7 > x x x > −3.73.7

✗

42 Chapter 1 Solving Equations and Inequalities

Write and solve an inequality that represents the value of x.

19. The base is less than or equal 20. The height is greater than to the height. the base.

22

x 12

15

x 3

Solve the inequality. Graph the solution.

21. 6m > − 54 22. z —

6 < 8 23.

v —

2 ≤ − 15

24. 51 ≤ 17c 25. 7

— 10

x < − 3

— 5

26. − 12.4 ≥ h

— 5

27. g —

5.1 > − 4 28. 28.8 < 3.2d 29. 9.8b ≥ − 29.4

30. ERROR ANALYSIS Describe and correct the error in solving the inequality.

Solve the inequality. Graph the solution.

31. n

— − 4

< 5 32. 0 ≥ w

— − 8

33. 6 > b ÷ (− 3)

34. − 3p > 72 35. − 27 ≤ − 5.4a 36. u —

− 1.8 < − 2.5

37. − 0.5d ≥ − 3.4 38. h

— − 8

> 3

— 4

39. 21.6 ≤ − 7.2x

40. ERROR ANALYSIS Describe and correct the error in solving the inequality.

41. FUNDRAISER You are selling sandwiches as a fundraiser. Your goal is to raise at least $225.

a. Write and solve an inequality to determine how many sandwiches you must sell to meet your goal.

b. How does your answer to part (a) change when the price decreases? increases?

3x

—— 33

> −2727 ——

33

x x < −9

✗

m —

−5 ≤ 12 12

((−−5) 5) ⋅⋅ m — −5

≤ ((−−5) 5) ⋅⋅ 12 12

m ≤ −6060

✗

22

33

Sandwiches$4.50 each

Section 1.6 Solving One-Step Inequalities 43

Solve the equation. Check your solution. (Section 1.2)

54. 29 = 17 − 2x 55. 3

— 4

m − 1

— 4

m − 6 = 1

— 2

56. 3(2.5 − k) − k = 14.5

57. MULTIPLE CHOICE The inside diameter of the cooler is 1 foot. About how many gallons of water does the cooler contain? (1 ft3 ≈ 7.5 gal) (Skills Review Handbook)

○A 1 gal ○B 8 gal

○C 12 gal ○D 18 gal

Write and solve an inequality that represents the value of x.

42. Area < 30 m2 43. Area ≥ 108 mm2

12 m

x

x

9 mm

REASONING Determine whether the statement is always, sometimes, or never true. Explain your reasoning.

44. If k is greater than 0, then kx > 0.

45. If k is greater than 0 and x is greater than 0, then kx > 0.

46. If k is less than 0, then kx < 0.

47. If k is less than 0 and x is greater than 0, then kx > 0.

48. BOWLING You can rent bowling shoes each time you bowl or you can buy a new pair for $48. Write and solve an inequality to determine when it is less expensive to buy new bowling shoes than to rent.

49. LAUNDROMAT A dryer at a laundromat will run for 10 minutes on one quarter. To dry your clothes, you need to run the dryer for at least 50 minutes. How much will it cost to dry your clothes?

Let a > b and x > y. Tell whether the statement is

always true. Explain your reasoning.

50. a + x > b + y 51. a − x > b − y

52. ax > by 53. a

— x

> y —

b

> 0.

Rental fee: $2.50p

$0.25

18 in.