Chapter 3-Inequalities

Lesson 3.5Solving Compound Inequalities

Vocabulary

Compound Inequality› Two inequalities that are joined by the

word and or the word or

Writing a compound inequality › Let’s say you have the inequalities:

X ≥ -5 AND X ≤ 7

› You can join them together as a COMPOUND INEQUALITY by writing it this way: -5 ≤ X ≤ 7

Formula for Compound Inequality

Smaller Number

Inequality Sign

Variable Inequality Sign

Larger Number

Graphing Compound Inequalities

Use a number line to graph inequalities.› Graph -5 ≤ X ≤ 7 › The black shaded portion represents the

compound inequality above

-7 -6 -5 -4 -3 -2 –1 0 1 2 3 4 5 6 7 8

Write and Graph the following Compound Inequalities

All real numbers that are AT LEAST -2 and AT MOST 4.

x ≤ 4 and x ≥ -2 -2 ≤ x ≤ 4

Write and Graph the Compound Inequality

Temperatures that are ABOVE 32 degrees but NOT AS HIGH AS 40 degrees.

t > 32 and t < 4032 < t < 40

Solving a Compound Inequality Containing AND

Solve -4 < x – 5 ≤ -1

•First step:•Write the compound inequality as TWO inequalities joined by AND

-4 < x – 5 AND x – 5 ≤ -1

Solving Compound Inequalities Containing

AND Second Step:

› Solve each inequality as you normally would. -4 < x – 5 AND x – 5 ≤ -1

+5 + 5 +5 +5

1 < x AND x ≤ 4

Solution:1 < x ≤ 4

Examples:

-6 ≤ 3x < 15

-3 < 2x – 1 < 7

7 < -3x + 1 ≤ 13

Writing Compound Inequalities Containing OR

A solution of a compound inequality joined by the word or is any number that makes EITHER inequality true.

Example:› All real numbers that are less than -3 OR

greater than 7.

X < -3 or X > 7

Graphing Compound Inequalities Containing OR

Use a number line to graph inequalities.› Graph x < -3 or x > 7

-7 -6 -5 -4 -3 -2 –1 0 1 2 3 4 5 6 7 8

Solving a Compound Inequality Containing OR

Solve 4v + 3 < -5 OR -2v + 7 < 1

•First step:•Write the compound inequality as TWO inequalities joined by OR

4v + 3 < -5 OR -2v + 7 < 1

Solving Compound Inequalities Containing OR

Second Step: › Solve each inequality as you normally would.

4v + 3 < –5 OR -2v + 7 < 1

- 3 -3 - 7 -74v < -8 -2v < -6

÷ 4 ÷ 4 ÷ -2 ÷ -2 v < - 2 OR v > 3

- 2 3

How do I know which symbol to use in a word problem?

Look for these WORD CLUES to help you determine which inequality symbol to use.

> < ≥ ≤

• Is more than

•Is greater than

• Is larger than

• Above

• Is less than

•Is smaller than

• Below

•Minimum• At least• No less

than• No

smaller than

•Maximum• At most• No

greater than

• No more than