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≤2*x*≤8

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

4≤2*x*

≤2

2≤x

x≥2

≤822≤

x≤4

Graph:

Example 1

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

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

5≤ + 5

0≤+5 < 6

0≤x

< 10≤

x< 3

Graph:

Example 2

*Example 3*: Solve and graph:
3(*x* - 2) < 9
**or**
3(*x* - 2) > 15
(union of sets)

3(*x* - 2) < 9

3(x- 2 < 3

x< 5

x- 2 > 5

x> 7

Graph:

Example 3

*Example 4*: Solve and graph:
{*x* : 2*x*≤*x* - 3}∪{*x* : *x* < 3*x* - 4}

2*x*≤*x* - 3
or
*x* < 3*x* - 4
(union of sets)

2*x*≤*x* - 3

x≤ - 3

-2x< - 4

x>2

Graph:

Example 4