Compound inequalities refer to a group of multiple inequalities joined by either “and” or “or”. Venn diagrams are a good visual way to represent the solutions to the compound inequalities. Each circle in a Venn diagram represents the set of solutions for each individual inequality

**Compound Inequality: **Two or more inequalities joined by *and *or or.

Inequalities joined by *and*

- The solution must make
*both*inequalities true - If a number makes only one of the inequalities true, that number is not a solution for the compound inequality
- Solutions are like the intersection of two sets. A is the set of solutions for one of the inequalities, and B is the set of solutions for the other inequalities. The solution for the compound inequality is \(A \cap B\)

Inequalities joined by *or*

- The solution must make
*at least one*inequality true - Solutions are like the union of two sets. A is the set of solutions for one of the inequalities, and B is the set of solutions for the other inequalities. The solution for the compound inequality is \(A \cup B\)

\(x > a\) and \(x < b\) (can be rewritten as \(a < x < b\))

- \(a\) must be less than \(b\)

Example: \(x ≥ -40\) and \(x < 60\) (can be rewritten as \(-40 ≤ x < 60\))

\(x < a \text{ or } x > b \)

- a must be greater than b

Example: \(x ≤ - 1 \text{ or } x ≥ 4\)

Separate the inequalities and solve them separately.

- Review the
*Linear Inequalities*study guide on how to solve inequalities. - Combine the solutions at the end.

Example: \(3x-5 < x+9 ≤ 5x+13\)

\(3x-5 < x+9\)

\(2x < 14\)

\(x < 7\)

\(2x < 14\)

\(x < 7\)

and

\(x+9 ≤ 5x+13\)

\(-4 ≤ 4x\)

\(-1 ≤ x\)

\(-4 ≤ 4x\)

\(-1 ≤ x\)

Answer: \(x < 7\) and \(x ≥ -1\) (rewritten as \(-1 ≤ x < 7\))

Solve each inequality separately

- Review the
*Linear Inequalities*study guide on how to solve inequalities.

Example: \(9-2x ≤ 3 \text{ or } 3x+10 ≤ 6 - x\)

\(9-2x ≤ 3\)

\(-2x ≤ -6\)

\(x ≥ 3\)

\(-2x ≤ -6\)

\(x ≥ 3\)

or

\(3x+10 ≤ 6 - x\)

\(4x ≤ -4\)

\(x ≤ -1\)

\(4x ≤ -4\)

\(x ≤ -1\)

Answer: \(x ≥ 3\) and \(x ≤ -1\)