Graphing and Solving Inequalities Guide

Sep 7, 2024

Graphing and Solving Inequalities

Key Concepts

  • Representing inequalities on number lines
  • Shading directions based on inequality signs
  • Intervals and interval notation
  • Solving inequalities with fractions, variables, and absolute values

Graphing Inequalities on Number Lines

Example 1: x > 2

  • Open circle at 2 (no underline)
  • Shade to the right (greater than)
  • Interval Notation: (2, ∞)

Example 2: x ≥ -1

  • Closed circle at -1 (includes -1)
  • Shade to the right (greater than or equal to)
  • Interval Notation: [-1, ∞)

Example 3: x < 4

  • Open circle at 4
  • Shade to the left (less than)
  • Interval Notation: (-∞, 4)

Example 4: x ≤ -2

  • Closed circle at -2
  • Shade to the left (less than or equal to)
  • Interval Notation: (-∞, -2]

Example 5: 1 < x ≤ 4

  • Open circle at 1 and closed circle at 4
  • Shade between 1 and 4
  • Interval Notation: (1, 4]

Example 6: x < -2 or x ≥ 3

  • Open circle at -2 and closed circle at 3
  • Shade left of -2 and right of 3
  • Interval Notation: (-∞, -2) ∪ [3, ∞)

Solving Inequalities

Example 1: x + 4 > 5

  1. Subtract 4: x > 1
  2. Interval Notation: (1, ∞)

Example 2: -2x + 5 < 12

  1. Subtract 5: -2x < 7
  2. Divide by -2 (flip inequality): x > -3.5
  3. Interval Notation: (-3.5, ∞)

Example 3: 7 - 2x ≤ 12

  1. Subtract 7: -2x ≤ 5
  2. Divide by -2 (flip inequality): x ≥ -2.5
  3. Interval Notation: [-2.5, ∞)

Example 4: 4 - 2x ≥ 3x + 19

  1. Move terms: -5x ≥ 15
  2. Divide by -5 (flip inequality): x ≤ -3
  3. Interval Notation: (-∞, -3]

Example 5: 1/3x + 4 > 8

  1. Subtract 4: 1/3x > 4
  2. Multiply by 3: x > 12
  3. Interval Notation: (12, ∞)

Solving Inequalities with Absolute Value

Example 1: |x| < 4

  • Two equations: x < 4 and x > -4
  • Interval Notation: (-4, 4)

Example 2: |2x + 3| ≥ 8

  1. Two equations: 2x + 3 ≥ 8 and 2x + 3 ≤ -8
  2. Solve both:
    • 2x ≥ 5 → x ≥ 2.5
    • 2x ≤ -11 → x ≤ -5.5
  3. Interval Notation: (-∞, -5.5] ∪ [2.5, ∞)

Example 3: 5 - 3|4x + 1| ≥ -9

  1. Move terms: |4x + 1| ≤ 14/3
  2. Write two equations: 4x + 1 ≤ 14/3 and 4x + 1 ≥ -14/3
  3. Solve both:
    • 4x ≤ 11/3 → x ≤ 11/12
    • 4x ≥ -17/3 → x ≥ -17/12
  4. Interval Notation: [-17/12, 11/12]

Summary

  • Understand different types of inequalities and how to graph and solve them.
  • Be familiar with using interval notation to express results succinctly.
  • Practice solving inequalities in different forms including those with fractions and absolute values.