Solving linear inequalities

Linear inequalities are similar to linear equations, but instead of an equals sign (=), they use inequality signs.

Solving linear inequalities

Solving linear inequalities is very similar to solving linear equations. The main difference is that when we multiply or divide both sides of the inequality by a negative number, we need to reverse the direction of the inequality sign.

Example: Solve the inequality 3x + 5 < 14

• Subtract 5 from both sides:
  • 3x + 5 - 5 < 14 - 5
  • 3x < 9
• Divide both sides by 3:
  • \frac{3x}{3} < \frac{9}{3}
  • x < 3

Answer: x < 3

Example: Solve the inequality -2x + 4 \geq 10

• Subtract 4 from both sides:
  • -2x + 4 - 4 \geq 10 - 4
  • -2x \geq 6
• Divide both sides by -2 (remember to reverse the inequality sign):
  • \frac{-2x}{-2} \leq \frac{6}{-2}
  • x \leq -3

Answer: x \leq -3

Example: Solve the inequality 5x - 7 > 3x + 1

• Subtract 3x from both sides:
  • 5x - 3x - 7 > 3x - 3x + 1
  • 2x - 7 > 1
• Add 7 to both sides:
  • 2x - 7 + 7 > 1 + 7
  • 2x > 8
• Divide both sides by 2:
  • \frac{2x}{2} > \frac{8}{2}
  • x > 4

Answer: x > 4