Lesson 10: Solving Linear Inequalities

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

Steps to Solve Linear Inequalities

1. Treat the inequality like an equation: add, subtract, multiply, or divide on both sides as needed.
2. Important rule: If you multiply or divide both sides by a negative number, you must flip the inequality symbol.
3. Simplify the result and write the solution.
4. Graph the solution set on a number line.

Examples

Example 1: Simple Inequality

Solve: \(x + 3 < 7\)

Subtract 3 from both sides:

\(x < 4\)

Solution: all numbers less than 4.

Example 2: Multiplying by a Negative

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

Divide both sides by -2, and flip the inequality:

\(x \le -3\)

Solution: all numbers less than or equal to -3.

Example 3: Two-Step Inequality

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

1. Add 5 to both sides: \(3x > 9\)
2. Divide by 3: \(x > 3\)

Solution: all numbers greater than 3.

Graphing the Solutions

Just like in Lesson 9, use:

• Open circles for < or >
• Closed circles for ≤ or ≥
• Shade left for “less than,” right for “greater than”

Example: \(x \le -3\) → closed circle at -3, shading to the left.

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 linear equations, but remember the special rule: multiplying or dividing by a negative number flips the inequality sign. Solutions are shown on a number line with open or closed circles and shading to indicate the solution set.