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Understanding Graphing of Linear Inequalities

May 1, 2025

Math Antics: Graphing Inequalities in Algebra

Introduction

  • Presenter: Rob from Math Antics
  • Focus: Using inequality signs in algebraic equations (e.g., y = mx + b)
  • Differences between equations and inequalities

Basic Concepts

  • Inequalities: Use greater-than (>) and less-than (<) signs, sometimes with an equal sign (≄ or ≀).
  • Equation of a Line: y = mx + b
    • Example: y = x (m=1, b=0)
  • Transition to Inequality: Change y = x to y ≄ x

Graphing Basic Inequalities

  • Graphing y = x results in a diagonal line through quadrants I and III.
  • Inequality Change: y ≄ x allows y to be greater than x.
    • Points above the line make the inequality true.
    • Shade the area above the line to represent all solutions.

Conventions in Graphing

  • Solid Line: Used when equality is included (e.g., ≄ or ≀)
  • Dashed Line: Used when equality is not included (e.g., > or <)
  • Boundary Line: Serves as a border between solution and non-solution areas.

Graphing Linear Inequalities: Example

  • Example Inequality: y < 2x - 3
  • Steps:
    1. Graph Boundary Line:
      • Treat inequality as equation: y = 2x - 3
      • Plot points and draw a line (dashed for < or >, solid for ≀ or ≄)
    2. Pick a Test Point:
      • Choose a point not on the line (e.g., (0,0))
      • Plug into inequality to verify which side is in the solution set
    3. Shade the Correct Side:
      • If test point is a solution, shade its side
      • If not, shade the opposite side

Simplifying Inequalities

  • Equal Operations: Same principles as solving equations (e.g., combine like terms)
  • Switching Inequality Signs:
    • Necessary when reversing the sides of an inequality
    • Required when multiplying or dividing by a negative number

Solving Example

  • Example: Solve -2y + 1 > x + 5 for y
    1. Subtract 1 on both sides, becomes -2y > x + 4
    2. Divide by -2, flip inequality, becomes y < -x/2 - 2

Key Takeaways

  • Basic steps for graphing and simplifying inequalities.
  • Importance of flipping inequalities when certain operations are performed.
  • Encouragement to practice with problems and revisit concepts if needed.

Additional Resources

Full transcript