Linear Inequalities

Solving and graphing linear inequalities in one variable

Linear Inequalities

Inequality Symbols

  • << : less than
  • >> : greater than
  • ≤\leq : less than or equal to
  • ≥\geq : greater than or equal to

Solving Inequalities

Solve inequalities like equations, but reverse the inequality symbol when multiplying or dividing by a negative number.

Example 1: Solve x+5<12x + 5 < 12 x+5−5<12−5x + 5 - 5 < 12 - 5 x<7x < 7

Example 2: Solve −3x≥15-3x \geq 15 −3x−3≤15−3\frac{-3x}{-3} \leq \frac{15}{-3} (reverse the symbol!) x≤−5x \leq -5

Graphing on a Number Line

  • Open circle for << or >> (not included)
  • Closed circle for ≤\leq or ≥\geq (included)
  • Shade the region that satisfies the inequality

Compound Inequalities

AND: −3<x≤5-3 < x \leq 5 means x>−3x > -3 AND x≤5x \leq 5

OR: x<−2x < -2 OR x>3x > 3

📚 Practice Problems

1Problem 1easy

❓ Question:

Solve and graph: x−3≥7x - 3 \geq 7

💡 Show Solution

Step 1: Add 3 to both sides x−3+3≥7+3x - 3 + 3 \geq 7 + 3 x≥10x \geq 10

Graph: Draw a closed circle at 10 and shade to the right.

Answer: x≥10x \geq 10

2Problem 2medium

❓ Question:

Solve: −2x+5>13-2x + 5 > 13

💡 Show Solution

Step 1: Subtract 5 from both sides −2x>8-2x > 8

Step 2: Divide by -2 (REVERSE the inequality!) x<−4x < -4

Answer: x<−4x < -4

3Problem 3hard

❓ Question:

Solve: −5≤2x+1<9-5 \leq 2x + 1 < 9

💡 Show Solution

This is a compound inequality. Solve by working on all three parts:

Step 1: Subtract 1 from all parts −5−1≤2x+1−1<9−1-5 - 1 \leq 2x + 1 - 1 < 9 - 1 −6≤2x<8-6 \leq 2x < 8

Step 2: Divide all parts by 2 −3≤x<4-3 \leq x < 4

Answer: −3≤x<4-3 \leq x < 4

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