# Compound Inequalities

## Compound Inequalities

To solve a compound inequality, first separate it into two inequalities. Determine whether the answer should be a union of sets ("or") or an intersection of sets ("and"). Then, solve both inequalities and graph.

If it is unclear whether the inequality is a union of sets or an intersection of sets, then ##test each region## to see if it satisfies the compound inequality.

Example 1: Solve and graph: 4≤2x≤8

4≤2xand2x≤8 (intersection of sets)
4≤2x

2≤x
x≥2
2x≤8
≤82
x≤4
2≤x and x≤4.
Graph:
Example 1

Example 2: Solve and graph: {x : 5≤ +5 < 6}

5≤ + 5and +5 < 6 (intersection of sets)
5≤ + 5

0≤
0≤x
+5 < 6
< 1
x < 3
0≤x and x < 3.
Graph:
Example 2

Example 3: Solve and graph: 3(x - 2) < 9or3(x - 2) > 15 (union of sets)

3(x - 2) < 9

x - 2 < 3
x < 5
3(x - 2) > 15
x - 2 > 5
x > 7
x < 5 or x > 7.
Graph:
Example 3

Example 4: Solve and graph: {x : 2xx - 3}∪{x : x < 3x - 4}

2xx - 3 or x < 3x - 4 (union of sets)
2xx - 3

x≤ - 3
x < 3x - 4
-2x < - 4
x >2
x≤ - 3 or x > 2.
Graph:
Example 4