Algebra I

Compound Inequalities

Big Picture

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

Key Terms

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

Types of Compound Inequalities

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 \(A \cup B\)

Compound Inequalities on a Number Line

Inequality Joined by And

\(x > a\) 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 \(-40 ≤ x < 60\))

Inequality Joined by Or

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

  • a must be greater than b

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

Remember that an open circle means that point is not included and a filled circle means that point is included.

Solving Compound Inequalities

Inequality Joined by And

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: \(3x-5 < x+9 ≤ 5x+13\)

\(3x-5 < x+9\)
\(2x < 14\)
\(x < 7\)
and
\(x+9 ≤ 5x+13\)
\(-4 ≤ 4x\)
\(-1 ≤ x\)

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

Inequality Joined by Or

Solve each inequality separately

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

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

\(9-2x ≤ 3\)
\(-2x ≤ -6\)
\(x ≥ 3\)
or
\(3x+10 ≤ 6 - x\)
\(4x ≤ -4\)
\(x ≤ -1\)

Answer: \(x ≥ 3\) and \(x ≤ -1\)