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 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 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 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 < - 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 6: 2x - 3 < 5≤2 - 3x

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