1.7 Solving Two-Step Inequalities - Big Ideas Learning

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Spanish

1.7 Solving Two-Step Inequalities

STANDARDS OF LEARNING

8.15

How can you solve a two-step inequality?

1 ACTIVITY: Matching Inequalities

Work with a partner. Match the inequality with its graph.

a. 3x + 2 < 11

b. 3x - 4 5

c. --x - 1 0

2

d. 9 < 2x + 3

e. 5 4x - 7

f. --x + 6 < 8

2

A.

-8

-6

-4

-2

0

2

4

6

8

B.

-8

-6

-4

-2

0

2

4

6

8

C.

-8

-6

-4

-2

0

2

4

6

8

D.

-8

-6

-4

-2

0

2

4

6

8

E.

-8

-6

-4

-2

0

2

4

6

8

F.

-8

-6

-4

-2

0

2

4

6

8

2 ACTIVITY: Writing an Inequality

Work with a partner. One of your favorite stores is having a 75% off sale. You have $20. You want to buy a pair of jeans.

a. Which of the following represents your ability to buy the jeans with $20?

0.25x < 20

0.25x 20

0.25x > 20

0.25x 20

b. What does x represent? Graph the possible values of x on a number line.

c. Can you afford a pair of jeans that originally costs $100? Explain.

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Chapter 1 Solving Equations and Inequalities

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3 ACTIVITY: Spaceman Game

x + 13 90

x-2 5

39 < 3x

65 < x + 10

2x 22

x + 5 > 24

33 x - 5

3x > 126

6 < x-2

x+2 3

x + 30 84

35 x + 6

32 x - 29

7x > 441

110 < x + 32

180 4x

x - 52 32

x - 9 > 21

2x 178

4 < x+1

x - 24 > 34

58 < x - 28

134 < 2x

x --

45

2

17 < --x

3

x --

5

3

8

x --

2

--x 12

4

x --

> 14

5

37 --x

2

START

Play with a partner. Player 1: Start the game by finding the inequality from the list that has a

solution of x 1. This allows you to destroy Asteroids 1, 2, and 3. Player 2: Asteroid 3 is destroyed. Find the inequality from the list that has a

solution of x > 3 or x 4. This allows you to destroy Asteroid 4, an alien, and Asteroid 6. You can only destroy asteroids and aliens in a straight line. Each asteroid is worth 1 point. Each alien is worth 2 points. Take turns until someone reaches the end of the maze. The player with the most points wins. If an incorrect inequality is chosen, the player loses that turn.

FINISH

4. IN YOUR OWN WORDS How can you solve a two-step inequality?

Use what you learned about solving two-step inequalities to complete Exercises 3 and 4 on page 48.

Section 1.7 Solving Two-Step Inequalities

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1.7 Lesson

Lesson Tutorials

You can solve two-step inequalities in much the same way you solve two-step equations.

EXAMPLE 1 Solving Two-Step Inequalities

a. Solve 5x - 4 11. Graph the solution.

5x - 4 11

Write the inequality.

Step 1: Undo the subtraction.

+4 +4

Add 4 to each side.

5x 15

Simplify.

Step 2: Undo the multiplication.

5x --

15 --

5 5

x 3

Divide each side by 5. Simplify.

The solution is x 3.

x 3

-3 -2 -1 0 1 2 3

Check: x = 0 is not a solution.

4567

Check: x = 4 is a solution.

b. Solve -- y + 7 < 9. Graph the solution.

-6

Step 1: Undo the addition.

y -- -6

+

7

<

9

-7 -7

Write the inequality. Subtract 7 from each side.

y -- -6

<

2

Simplify.

Step 2: Undo the division.

-6 -- y > -6 2 -6

Multiply each side by -6. Reverse the inequality symbol.

y > -12

Simplify.

The solution is y > -12.

y > -12

-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2

Exercises 5?10

Solve the inequality. Graph the solution.

1. 4b - 1 < 7

2. 8 + 9c -28

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Chapter 1 Solving Equations and Inequalities

3. -- n + 11 > 12

-2

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Spanish

EXAMPLE 2 Standardized Test Practice

Which graph represents the solution of -7(x + 3) 28?

A

-10 -9 -8 -7 -6 -5 -4

B

-10 -9 -8 -7 -6 -5 -4

C

4 5 6 7 8 9 10

D

4 5 6 7 8 9 10

-7(x + 3) 28

-7x - 21 28

Step 1: Undo the subtraction.

+ 21 + 21

-7x 49

Step 2: Undo the multiplication.

-- -7x -- 49

-7 -7

x -7

Write the inequality. Use Distributive Property. Add 21 to each side. Simplify. Divide each side by -7. Reverse the inequality symbol. Simplify.

The correct answer is B .

EXAMPLE 3 Real-Life Application

Remember

The mean in Example 3 is equal to the sum of the game scores divided by the number of games.

You need a mean score of at least 90 to advance to the next round of the trivia game. What score do you need on the fifth game to advance?

Use the definition of mean to write and solve an inequality. Let x be the

score on the fifth game.

95 + 91 + 77 + 89 + x ----

90

5

The phrase "at least" means greater than or equal to.

352 + x --

90

5

Simplify.

5 -- 352 + x 5 90 5

Multiply each side by 5.

352 + x 450

Simplify.

- 352

- 352

Subtract 352 from each side.

x 98

Simplify.

You need at least 98 points to advance to the next round.

Exercises 12?17

Solve the inequality. Graph the solution.

4. 2(k - 5) < 6

5. -4(n - 10) < 32

6. -3 0.5(8 + y)

7. WHAT IF? In Example 3, you need a mean score of at least 88 to advance to the next round of the trivia game. What score do you need on the fifth game to advance?

Section 1.7 Solving Two-Step Inequalities

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1.7 Exercises

Help with Homework

1. WRITING Compare and contrast solving two-step inequalities and solving two-step equations.

2. OPEN-ENDED Describe how to solve the inequality 3(a + 5) < 9.

93++4(-+(6-9(3)-=+)9=3()-=1)=

Match the inequality with its graph. 3. --t - 1 -3

3

A.

9 8 7 6 5 4 3 2

4. 5x + 7 32 A.

23456789

B. 9 8 7 6 5 4 3 2

B.

23456789

C.

C.

9 8 7 6 5 4 3 2

Solve the inequality. Graph the solution.

1 5. 7b + 4 11

6. 2v - 4 < 8

8. --4 < 3w - -- 11

5

5

9. 1.8 < 0.5 - 1.3p

23456789

7.

1

-

m --

6

3

10. -2.4r + 9.6 4.8

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

--x + 6 3

4

x + 6 12

x 6

Solve the inequality. Graph the solution.

2 12. 6( g + 2) 18

13. 2( y - 5) 16

15. ---1(u + 2) > 5

3

16. 2.7 > 0.9(n - 1.7)

14. -10 --5(h - 3)

3

17. 10 > -2.5(z - 3.1)

18. ATM Write and solve an inequality that represents the number of $20 bills you can withdraw from the account without going below the minimum balance.

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Chapter 1 Solving Equations and Inequalities

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