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Understanding Linear Inequalities and Solutions

Jan 25, 2025

Lecture Notes: Linear Inequalities

Introduction

  • Discussion on linear inequalities.
  • Emphasis on a detailed review.
  • Key Rule: If you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.

Example 1: Solving 2x + 5 < 17

  • Steps to Solve:
    • Subtract 5 from both sides: 17 - 5 = 12.
    • Divide by 2: 12 / 2 = 6.
  • Graphical Representation:
    • Use a number line from -∞ to ∞.
    • For x < 6, place an open circle at 6 and shade to the left.
  • Interval Notation:
    • Solutions are from -∞ to 6, written as (-∞, 6).

Key Concepts

  • Inequality Solution:
    • Solve like a regular equation.
    • Reverse the inequality when multiplying/dividing by a negative.
  • Graph and Interval Notation:
    • Draw the number line.
    • Write solutions in interval notation.

Example 2: Solving -5x ≤ 30

  • Steps to Solve:
    • Divide by -5, reverse inequality: x ≥ -6.
  • Graphical Representation:
    • Number line from -∞ to ∞.
    • Closed circle at -6, shade to the right.
  • Interval Notation:

Special Cases

  • True Statement Example 1 < 7:
    • Inequality true for all real numbers.
    • Number line shaded entirely.
    • Interval notation: (-∞, ∞).
  • No Solution Example 6 ≥ 11:
    • False statement, thus no solution.
    • Number line remains unshaded.
    • No interval notation.

Additional Examples

Example 3: x - 4/6 < x - 2/9 + 5/18

  • Common Denominator Approach:
    • Multiply terms to achieve common denominator.
    • Solve the inequality by working on numerators.
  • Solution and Graphing:
    • x < 13, open circle at 13, shade left.
  • Interval Notation:
    • (-∞, 13).

Example 4: Compound Inequalities

  • Example: 2 < x - 7 < 5
  • Solution Steps:
    • Add 7 to each part: 9 < x < 12.
  • Graphical Representation:
    • Open circles at 9 and 12, shade between.
  • Interval Notation:
    • (9, 12).

Summary

  • Understanding linear inequalities involves solving, graphing, and using interval notation.
  • Watch out for multiplication/division by negative numbers to ensure the correct direction of inequalities.
  • Compound inequalities indicate a range of values.

These notes cover the fundamental approach to solving linear inequalities along with their graphical and interval notation representations.

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