# 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

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≤ + 5 and +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) < 9 or 3(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 : 2xx - 3}∪{x : x < 3x - 4}

2xx - 3 or x < 3x - 4 (union of sets)
2xx - 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 < - 2 or 3(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≥x x≤ - 1
x < 4 and x≤ - 1 .
Graph:
Example 6

## Take a Study Break

### Star Trek gets SEXY

Chris Pine and Zoe Saldana heat up the red carpet!

### Are you afraid of relationships?

Auntie SparkNotes can help!

### Sexy starlet style

See every single look from the Met Gala!

### Geeky Actors: Then and Now

Travel back in time!

### Villains We Want These Actresses to Play

From super cute to super bad!

### 10 Movies Better Than Their Books

What do you think?

### How To Look Like J-Law...

When you don't look like J-Law.

### 12 Scientific Inaccuracies in Into Darkness

What did Star Trek get wrong?

## The Book

### Read What You Love, Anywhere You Like

Get Our FREE NOOK Reading Apps