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
2x≤8≤
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2≤x
x≥2
2≤x and x≤4 .≤82
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
0≤x and x < 3 .< 1
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 < 33(x - 2) > 15
x < 5
x - 2 > 5x < 5 or x > 7 .
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≤ - 3x < 3x - 4
-2x < - 4x≤ - 3 or x > 2 .
x >2
Graph:
Example 5: Solve and graph:
2x - 2 < - 2
or
3(x + 5) > 2x + 15
(union of sets)
2x - 2 < - 2
2x < 03(x + 5) > 2x + 15
x < 0
3x + 15 > 2x + 15x < 0 or x > 0 .
3x > 2x
x > 0
Graph:
Example 6:
2x - 3 < 5≤2 - 3x
2x - 3 < 5
and
5≤2 - 3x
(intersection of sets)
2x - 3 < 5
2x < 85≤2 - 3x
x < 4
3≤ - 3x -1≥x x≤ - 1x < 4 and x≤ - 1 .
Graph:
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