2 Solving Linear Inequalities - Big Ideas Learning

2 Solving Linear Inequalities

2.1 Writing and Graphing Inequalities 2.2 Solving Inequalities Using Addition or Subtraction 2.3 Solving Inequalities Using Multiplication or Division 2.4 Solving Multi-Step Inequalities 2.5 Solving Compound Inequalities

Withdraw Money (p. 71) Digital Camera (p. 62)

Mountain Plant Life (p. 77) SEE the Big Idea

Microwave Electricity (p. 56)

Natural Arch (p. 51)

Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems

arising in everyday life, society, and the workplace.

Maintaining Mathematical Proficiency

Graphing Numbers on a Number Line (6.N2.SC.)C.6c)

Example 1 Graph each number.

a. 3 b. -5

-5 -4 -3 -2 -1 0 1 2 3 4 5

b. -5 Example 2 Graph each number.

a. 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 Example 2 Gb.raph-ea2ch number.

Graph thea.num4ber.

1. 6

4. 2 + -2

-5 -4 -3

2. 2

5. 1 - -4

-2 -1 0 1

The absolute value of a positive number is positive.

3. -1

6. -5 + 3

2345

Comparibn. g R-e2a l Numbers (6.NTSh.eCa.7baso)lute value of a negative number is positive.

Example 3 Complete the statement -1 -5 with , or =.

-5 -4 -3 -2 -1 0 1 2 3 4 5

Graph the nu-m1bisetor.the right of -5. So, -1 > -5.

1. 6

2. 2

Ex4a.m2p+le 4-2Evaluate 15 ? (-3). 5. 1 - -4

3. -1 6. -5 + 3

Comparing Real Numbers (6.2.C)

15 ? (-3) = -5

Example 3 Complete the statement -1

-5 with , or =.

Graph ?5.

Multiply or divide.

Graph ? 1.

10. -3(8)

11. -7 (-9)

12. 4 (-7)

-7 -6 -5 -4 -3 -2 -1 0 1 2 3

13. -24 ? (-6)-1 is to the right1o4f.--5.1S6o?, -21 > -5.

16. 6 8

17. 36 ? 6

Complete the statement with , or =.

15. 12 ? (-3) 18. -3(-4)

19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers, 7. (2c) mul9tiplying integers, and (d)8d. iv-id6ing inte5gers. Give an example9o.f e-ac1h2. -4

10. -7 -13

11. -8

8

12. -10

-18

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?

43

Mathematical Thinking

Mathematically proficient students select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (A.1.C)

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 x + 2.

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

Y1=2X-1 -2 d. x 10

Section 2.1 Writing and Graphing Inequalities

45

2.1 Lesson

Core Vocabulary

inequality, p. 46 solution of an inequality, p. 47 solution set, p. 47 graph of an inequality, p. 48 Previous expression

What You Will Learn

Write linear inequalities. Sketch the graphs of linear inequalities. Write linear inequalities from graphs.

Writing Linear Inequalities

An inequality is a mathematical sentence that compares expressions. An inequality contains the symbol , , or . To write an inequality, look for the following phrases to determine what inequality symbol to use.

Symbol

Key Phrases

<

is less than

is fewer than

Inequality Symbols

>

is greater is less than or is greater than

than

equal to

or equal to

is more than

is at most

is at least

is no more than is no less than

READING

The inequality 3 < n + 5 is the same as n + 5 > 3.

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

-2

An inequality is w - 3.5 -2.

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

3

<

n + 5

An inequality is 3 < n + 5.

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

0

An inequality is 0 2x + 1.

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. 2. ---170 is at least twice a number k minus 4.

46

Chapter 2 Solving Linear Inequalities

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