Compound Inequalities Study Guide

Algebra I

Compound Inequalities

Big Picture

Compound inequalities refer to a group of multiple inequalities joined by either “and” or “or”. Venn diagrams are a good visual way to represent the solutions to the compound inequalities. Each circle in a Venn diagram represents the set of solutions for each individual inequality

Key Terms

Compound Inequality: Two or more inequalities joined by and or or.

Types of Compound Inequalities

Inequalities joined by and

The solution must make both inequalities true
If a number makes only one of the inequalities true, that number is not a solution for the compound inequality
Solutions are like the intersection of two sets. A is the set of solutions for one of the inequalities, and B is the set of solutions for the other inequalities. The solution for the compound inequality is $A \cap B$

Inequalities joined by or

The solution must make at least one inequality true
Solutions are like the union of two sets. A is the set of solutions for one of the inequalities, and B is the set of solutions for the other inequalities. The solution for the compound inequality is $A \cup B$

Compound Inequalities on a Number Line

Inequality Joined by And

$x > a$ and $x < b$ (can be rewritten as $a < x < b$ )

$a$ must be less than $b$

Example: $x \geq - 40$ and $x < 60$ (can be rewritten as $- 40 \leq x < 60$ )

Inequality Joined by Or

$x < a or x > b$

a must be greater than b

Example: $x \leq - 1 or x \geq 4$

Remember that an open circle means that point is not included and a filled circle means that point is included.

Solving Compound Inequalities

Inequality Joined by And

Separate the inequalities and solve them separately.

Review the Linear Inequalities study guide on how to solve inequalities.
Combine the solutions at the end.

Example: $3 x - 5 < x + 9 \leq 5 x + 13$

3 x - 5 < x + 9

2 x < 14

x < 7

and

x + 9 \leq 5 x + 13

- 4 \leq 4 x

- 1 \leq x

Answer: $x < 7$ and $x \geq - 1$ (rewritten as $- 1 \leq x < 7$ )

Inequality Joined by Or

Solve each inequality separately

Review the Linear Inequalities study guide on how to solve inequalities.

Example: $9 - 2 x \leq 3 or 3 x + 10 \leq 6 - x$

9 - 2 x \leq 3

- 2 x \leq - 6

x \geq 3

3 x + 10 \leq 6 - x

4 x \leq - 4

x \leq - 1

Answer: $x \geq 3$ and $x \leq - 1$