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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 : 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 4


Example 5: Solve and graph: 2x - 2 < - 2or3(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 5


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≥xx≤ - 1
x < 4 and x≤ - 1.
Graph:
Example 6

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