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

≤82
x≤4

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

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

x - 2 > 5
x > 7

Example 3

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

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

x≤ - 3

-2x < - 4
x >2

Example 4