2 Solving Linear Inequalities

2 Solving Linear Inequalities

2.1

2.2

2.3

2.4

2.5

2.6

Writing and Graphing Inequalities

Solving Inequalities Using Addition or Subtraction

Solving Inequalities Using Multiplication or Division

Solving Multi-Step Inequalities

Solving Compound Inequalities

Solving Absolute Value Inequalities

Camel Physiology (p. 91)

Mountain Plant Life (p. 85)

SEE the Big Idea

Digital Camera (p. 70)

Microwave Electricity (p. 64)

Natural Arch (p. 59)

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Maintaining Mathematical Proficiency

Graphing Numbers on a Number Line

(6.NS.C.6c)

Example 1 Graph each number.

a.

a.

3

3

b.

?5

b.

?5

?5

?4

?3

?2

?1

0

1

2

3

4

5

?2

?1

0

1

2

3

4

5

Example 2 Graph each number.

?5

?4

¨O4¨O

a.

?3

Example 2 Graph

each

¨O number.

b. ¨O ?2

a. ¨O 4 ¨O

The absolute value of a positive number is positive.

Graph the number.

1. 6

2.

4. 2 + ¨O ?2 ¨O

?5

?4

?3

?2

¨O2¨O

3.

?1 1 ?0¨O ?4 1

¨O

5.

¨O ?2 ¨O

b.

2

3

4

5

¨O ?1 ¨O

6. ?5 + ¨O 3 ¨O

The absolute value of a negative number is positive.

Comparing Real Numbers

(6.NS.C.7a)

Example 3

0

Complete

the statement ?1

?5 ?4 ?3 ?2 ?1

?5

with

,

or 4=.

1

2

3

5

Graph the number.

?1is to the right of ?5. So, ?1 > ?5.

2. ¨O 2 ¨O

1. 6

3.

4. 2 + ¨O ?2 ¨O

5. 1 ? ¨O ?4 ¨O

Example 4 Evaluate 15 ¡Â (?3).

¨O ?1 ¨O

6. ?5 + ¨O 3 ¨O

Comparing Real Numbers

Example 3 Complete the statement ?1

?5 with , or =.

15 ¡Â (?3) = ?5

Graph ¨C 5.

Multiply or divide.

10. ?3 (8)

?7

13. ?24 ¡Â (?6)

?6

?5

?4

Graph ¨C 1.

?3

?2

?1

0

? ?1 > ?5.

?1 is to the right of ?5. So,

1

2

3

? (?7)

11. ?7 (?9)

12. 4

14. ?16 ¡Â 2

15. 12 ¡Â (?3)

Complete

the statement with

16. 6 8

17. 36,

6 or =.

?

18. ?3(?4)

7. 2

9

8. ?6

5

9. ?12

?4

19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,

¨O ?8 ¨O integers.

¨O 8 ¨O Give an example12.of each.

10. (c)

?7multiplying

?13 integers, and (d)

11.dividing

?10

¨O ?18 ¨O

13. ABSTRACT REASONING A number a is to the left of a number b on the number line.

How do the numbers ?a and ?b compare?

Dynamic Solutions available at

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51

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Mathematical

Practices

Mathematically proficient students use technology tools to

explore concepts.

Using a Graphing Calculator

Core Concept

Solving an Inequality in One Variable

You can use a graphing calculator to solve an inequality.

1.

Enter the inequality into a graphing calculator.

2.

Graph the inequality.

3.

Use the graph to write the solution.

Using a Graphing Calculator

Use a graphing calculator to solve (a) 2x ? 1 < x + 2 and (b) 2x ? 1 ¡Ü x + 2.

SOLUTION

a.

Enter the inequality 2x ? 1 < x + 2 into a graphing calculator. Press graph.

Y1=2X-1

¡Ý

¡ñ

is less

than

¡ñ

is greater

than

¡ñ

is less than or

equal to

¡ñ

is greater than

or equal to

¡ñ

is fewer

than

¡ñ

is more

than

¡ñ

is at most

¡ñ

is at least

¡ñ

is no more than

¡ñ

is no less than

Writing Inequalities

Write each sentence as an inequality.

a. A number w minus 3.5 is less than or equal to ?2.

b. Three is less than a number n plus 5.

c. Zero is greater than or equal to twice a number x plus 1.

SOLUTION

a. A number w minus 3.5 is less than or equal to ?2.

w ? 3.5

¡Ü

?2

An inequality is w ? 3.5 ¡Ü ?2.

READING

b. Three is less than a number n plus 5.

The inequality 3 < n + 5 is

the same as n + 5 > 3.

3

n+5

<

An inequality is 3 < n + 5.

c. Zero is greater than or equal to twice a number x plus 1.

¡Ý

0

2x + 1

An inequality is 0 ¡Ý 2x + 1.

Monitoring Progress

Help in English and Spanish at

Write the sentence as an inequality.

1. A number b is fewer than 30.4.

7

2. ?¡ª

10 is at least twice a number k minus 4.

54

Chapter 2

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Solving Linear Inequalities

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