Algebra 101 - Unit 2

Lesson 10: Solving Linear Inequalities

Solving a linear inequality is very similar to solving a linear equation. The goal is to isolate the variable, but instead of one solution, inequalities usually have many solutions represented on a number line.

Steps to Solve Linear Inequalities

1. Solve like an equation: add, subtract, multiply, or divide on both sides.
2. Rule: If you multiply or divide by a negative number, flip the inequality symbol.
3. Simplify and write the solution.
4. Graph the solution on a number line.

Examples

Example 1: Simple Inequality

Solve: \(x + 3 < 7\)

\(x < 4\)

Example 2: Multiplying by a Negative

Solve: \(-2x \ge 6\)

\(x \le -3\)

Example 3: Two-Step Inequality

Solve: \(3x - 5 > 4\)

\(x > 3\)

Practice Problems

1. Solve and graph: \(x - 5 \ge 2\)
2. Solve and graph: \(-4x < 20\)
3. Solve and graph: \(2x + 7 \le 13\)
4. Solve and graph: \(-3x - 2 > 7\)

Lesson Summary

Solving linear inequalities is almost the same as solving equations, but remember: multiplying or dividing by a negative number flips the inequality sign. Solutions are graphed with open or closed circles and shaded regions on a number line.