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

≤
2≤x
x≥2

2x≤8

≤82
x≤4

2≤x and x≤4 .
Graph:

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

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

0≤
0≤x

+5 < 6

< 1
x < 3

0≤x and x < 3 .
Graph:

Example 3: Solve and graph: 3(x - 2) < 9 or 3(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 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

x < 3x - 4

-2x < - 4
x >2

x≤ - 3 or x > 2 .
Graph:

Example 5: Solve and graph: 2x - 2 < - 2 or 3(x + 5) > 2x + 15 (union of sets)

2x - 2 < - 2

2x < 0
x < 0

3(x + 5) > 2x + 15

3x + 15 > 2x + 15
3x > 2x
x > 0

x < 0 or x > 0 .
Graph:

Example 6: 2x - 3 < 5≤2 - 3x

2x - 3 < 5 and 5≤2 - 3x (intersection of sets)
2x - 3 < 5

2x < 8
x < 4

5≤2 - 3x

3≤ - 3x -1≥x x≤ - 1

x < 4 and x≤ - 1 .
Graph: