1. Objectives Solve compound inequalities with one variable.
Graph solution sets of compound inequalities with one variable.

3. Example 2A: Solving Compound Inequalities Involving AND
Solve the compound inequality and graph the solutions.
–5 < x + 1 < 2
Since 1 is added to x, subtract 1 from each part of the inequality.
–5 < x + 1 < 2
– – 1 – 1
–6 < x < 1
Graph –6 < x.
Graph x < 1.
Graph the intersection by finding where the two graphs overlap.
–10 –8 –6 –4 –2 2 4 6 8 10

4. Example 2B: Solving Compound Inequalities Involving AND
Solve the compound inequality and graph the solutions.
8 < 3x – 1 ≤ 11
8 < 3x – 1 ≤ 11
9 < 3x ≤ 12
Since 1 is subtracted from 3x, add 1 to each part of the inequality.
Since x is multiplied by 3, divide each part of the inequality by 3 to undo the multiplication.
3 < x ≤ 4

5. Example 3A: Solving Compound Inequalities Involving OR
Solve the inequality and graph the solutions.
8 + t ≥ 7 OR 8 + t < 2
8 + t ≥ 7 OR 8 + t < 2
Solve each simple inequality.
– –8 – −8
t ≥ –1 OR t < –6
Graph t ≥ –1.
Graph t < –6.
Graph the union by combining the regions.
–10 –8 –6 –4 –2 2 4 6 8 10

6. Example 3B: Solving Compound Inequalities Involving OR
Solve the inequality and graph the solutions.
4x ≤ 20 OR 3x > 21
4x ≤ 20 OR 3x > 21
x ≤ 5 OR x > 7
Solve each simple inequality.
Graph x ≤ 5.
Graph x > 7.
Graph the union by combining the regions.
–10 –8 –6 –4 –2 2 4 6 8 10

7. Solve the compound inequality and graph the solutions.
Check It Out! Example 2b
Solve the compound inequality and graph the solutions.
–4 ≤ 3n + 5 < 11
–4 ≤ 3n + 5 < 11
– – 5 – 5
–9 ≤ 3n <
Since 5 is added to 3n, subtract 5 from each part of the inequality.
Since n is multiplied by 3, divide each part of the inequality by 3 to undo the multiplication.
–3 ≤ n < 2
Graph –3 ≤ n.
Graph n < 2.
Graph the intersection by finding where the two graphs overlap. –5 –4 –3 –2 –1 1 2 3 4 5

8. Example 4A: Writing a Compound Inequality from a Graph
Write the compound inequality shown by the graph.
The shaded portion of the graph is not between two values, so the compound inequality involves OR.
On the left, the graph shows an arrow pointing left, so use either < or ≤. The solid circle at –8 means –8 is a solution so use ≤.
x ≤ –8
On the right, the graph shows an arrow pointing right, so use either > or ≥. The empty circle at 0 means that 0 is not a solution, so use >.
x > 0
The compound inequality is x ≤ –8 OR x > 0.

9. Example 4B: Writing a Compound Inequality from a Graph
Write the compound inequality shown by the graph.
The shaded portion of the graph is between the values –2 and 5, so the compound inequality involves AND.
The shaded values are on the right of –2, so use > or ≥. The empty circle at –2 means –2 is not a solution, so use >.
m > –2
The shaded values are to the left of 5, so use < or ≤. The empty circle at 5 means that 5 is not a solution so use <.
m < 5
The compound inequality is m > –2 AND m < 5 (or -2 < m < 5).

10. Check It Out! Example 4a
Write the compound inequality shown by the graph.
The shaded portion of the graph is between the values –9 and –2, so the compound inequality involves AND.
The shaded values are on the right of –9, so use > or . The empty circle at –9 means –9 is not a solution, so use >.
x > –9
The shaded values are to the left of –2, so use < or ≤. The empty circle at –2 means that –2 is not a solution so use <.
x < –2
The compound inequality is –9 < x AND x < –2
(or –9 < x < –2).

11. Check It Out! Example 4b
Write the compound inequality shown by the graph.
The shaded portion of the graph is not between two values, so the compound inequality involves OR.
On the left, the graph shows an arrow pointing left, so use either < or ≤. The solid circle at –3 means –3 is a solution, so use ≤.
x ≤ –3
On the right, the graph shows an arrow pointing right, so use either > or ≥. The solid circle at 2 means that 2 is a solution, so use ≥.
x ≥ 2
The compound inequality is x ≤ –3 OR x ≥ 2.