Study Guide: Solving Compound Inequalities

Study Guide
Solving Compound Inequalities

A compound inequality consists of two inequalities that are connect by the words and or or.

A compound inequality containing and is true only if both inequalities are true. Its graph is the intersection of the graphs of the two inequalities.
A compound inequality containing or is true if one or more of the inequalities is true. Its graph is the union of the two inequalities.

Example 1: x > −3 and x ≤ 4

The solution set, shown in the bottom graph, is {x| -3 < x ≤ 4}

Remember that open circles denote < and > while solid circles denote ≤ and ≥.

Example 2: t ≥ 8 or t < 5.

The solution set is {t| t ≥ 8 or t < 5}

Sometimes it is better to first solve each inequality and then graph the solution. Study the examples below.

Example 3: −3 ≤ p − 5 < 2

Both "sides" of this compound inequality must be true, so the solution is their intersection.

−3 ≤ p − 5

−3 + 5 ≤ p − 5 + 5

2 ≤ p

p − 5 < 2

p − 5 + 5 < 2 + 5

p < 7

The solution set is {p| 2 ≤ p < 7}

Example 4: 2a + 1 < 11 or a > 3a + 2

The solution to this compound inequality is the union of the solutions to each "side."

2a + 1 < 11

2a + 1 − 1 < 11 − 1

2a < 10

a < 5

a > 3a + 2

a − 3a > 3a − 3a + 2

−2a > 2

−2a ÷ −2 > 2 ÷ −2

a < −1

The solution set is {a| a < 5}