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Section 3.4

Compound Inequalities

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Objectives

• Basic Concepts

• Symbolic Solutions and Number Lines

• Numerical and Graphical Solutions

• Interval Notation

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Basic Concepts

A compound inequality consists of two inequalities joined by the words and or or.

2x > –5 and 2x ≤ 8

x + 3 ≥ 4 or x – 2 < –6

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Example

Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4

Solutionx + 2 < 7 and 2x – 3 > 3

Substitute 4 into the given compound inequality.4 + 2 < 7 and 2(4) – 3 > 3 6 < 7 and 5 > 3 True and True

Both inequalities are true, so 4 is a solution.

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Example (cont)

Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4

Solutionx + 2 < 7 and 2x – 3 > 3

Substitute –4 into the given compound inequality. –4 + 2 < 7 and 2(–4) – 3 > 3 – 2 < 7 and –11 > 3 True and False

To be a solution both inequalities must be true, so –4 is not a solution.

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Symbolic Solutions and Number Lines

We can use a number line to graph solutions to compound inequalities, such as x < 7 and x > –3.

x < 7

x > –3

x < 7 and x > –3

Note: A bracket, either [ or ] or a closed circle is used when an inequality contains ≤ or ≥. A parenthesis, either ( or ), or an open circle is used when an inequality contains < or >.

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Example

Solve 3x + 6 > 12 and 5 – x < 11 . Graph the solution.

Solution3x + 6 > 12 and 5 – x < 11

3 6x 6x

2x 6x

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Example

Solve each inequality. Graph each solution set. Write the solution in set-builder notation. a. b. c.Solutiona. b.

6 2 10w 4 4 8y 4 2

53 3

w

6 2 2 22 10w 4 8w

6 2 10w

| 4 8w w

1 2y

4 4 8y

| 1 2 y y

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Example (cont)

c.

4 25

3 3

w

4 2 5

3 3

w

4 23 5

33

33

w

4 2 15w

24 2 22 15w

6 13w

( 6) ( 11 ) 31 1w

6 13w 13 6w

| 13 6 w w

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Example

Solve x + 3 < –2 or x + 3 > 2

Solutionx + 3 < –2 or x + 3 > 2 x < –5 or x > –1

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Example

Write each expression in interval notation. a. –3 ≤ x < 7

b. x ≥ 4

c. x < –3 or x ≥ 5

d. {x|x > 0 and x ≤ 5}

e. {x|x ≤ 2 or x ≥ 5}

3,7

4,

, 3 5,

0,5

, 2 5,

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Example

Solve 2x + 3 ≤ –3 or 2x + 3 ≥ 5

Solution

2x + 3 ≤ –3 or 2x + 3 ≥ 5

2x ≤ –6 or 2x ≥ 2

x ≤ –3 or x ≥ 1

The solution set may be written as (, 3] [1, )