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## 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.

1. a ≤ 20 and a > 12
2. -6 < b < 14
3. c ≥ 10 or c ≤ 2
4. 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.)

## Try It

Draw the graph of each inequality.

1)  f ≤ 6 and f > -1

2)  g > -2 or g ≤ 7

3)  0 ≤ h ≤ 5

4)  j ≥ 44 or j < 15

Scroll Down for Answers…

1)

2)

3)

4)