1. The inequalities you have seen so far are simple inequalities
The inequalities you have seen so far are simple inequalities. When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality.

2. You can graph the solutions of a compound inequality involving AND by using the idea of an overlapping region. The overlapping region is called the intersection and shows the numbers that are solutions of both inequalities.

3. You can graph the solutions of a compound inequality involving OR by using the idea of combining regions. The combine regions are called the union and show the numbers that are solutions of either inequality.
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5. 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

6. 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

7. 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

8. 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

9. 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.

10. 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).