## Graph Compound Inequalities

## Introduction

A simple inequality is one that has a single critical number. This lesson goes beyond simple inequalities and into compound inequalities, which have 2 or more critical numbers. One major difference of a simple inequality and a compound inequality is that the graphs of compound inequalities have two individual dots.

The compound inequality above is just a pair of simple inequalities connected with the word “or.” This is a standard way to express a compound inequality.

## Lesson

The phrase*compound inequality*generally means a pair of inequalities joined by the word “and” or the word “or.” Here is one example of each:

When a compound inequality is connected by the word* and*, its solutions must satisfy both of the individual inequalities. In example 1, focus on how the solution comes from the two individual inequalities.

**Example 1****:** Graph the solution of the inequality x ≥ 3 and x < 7.

Note that one circle is open and the other is closed. These circles are determined by the simple inequalities that they correspond to. The critical number 3 has a closed circle from the inequality x ≥ 3 while the open circle on the 7 comes from x < 7. Inequalities that contain an “and” can also be written as a single inequality. In example 1, this inequality would be 3 ≤ x < 7.

When a compound inequality is connected by the word *or*, its solutions are the solutions to either of the two individual inequalities. Use example 2 to compare the solution to the individual inequalities that it comes from.

**Example 2****:** Graph the solution of the inequality y > 5 or y ≤ -2

Example 3 has several examples of graphing inequalities.

**Example 3****: ** Graph the solutions the inequalities.

- a ≤ 20 and a > 12
- -6 < b < 14
- c ≥ 10 or c ≤ 2
- c ≥ 10 and c ≤ 2

The graphs of a, b, and c are typical of compound inequalities. Part d looks similar to part c except for the word “and.” It is not possible for a number to be both greater than 10 and less than 2 at the same time, so there are no solutions at all and the graph should be left blank (while also stating in words that there are no solutions.)

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