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
≤
+5 < 6}
