Inequalities & Graphing Essentials

Inequality Symbols & Meanings

  • > : greater than
  • \ge : greater than or equal to (solid dot / line)
  • < : less than
  • \le : less than or equal to (solid dot / line)

Solving One-Variable Inequalities

  • Treat like equations: add / subtract / multiply / divide both sides.
  • Key rule: when multiplying or dividing by a negative, flip the inequality sign.
    • Example: -3x > 9 \;\Rightarrow\; x < -3
  • Express solution on a number line or in interval notation.
    • Open circle for > or <; closed circle for \ge or \le.

Compound Inequalities ("and" / "or")

  • "And" (written as a double inequality) ⇒ intersection of solutions.
    • Example: 5 \le 2x-3 < 13
    1. +3:\; 8 \le 2x < 16
    2. \div2:\; 4 \le x < 8 ⇒ interval [4,8)
  • "Or" inequalities describe values that satisfy either condition; graph as two separate rays.

Linear Inequalities in Two Variables (e.g., y > mx + b)

  • Step 1: Graph the boundary line y = mx + b.
    • Dashed line for > or <.
    • Solid line for \ge or \le.
  • Step 2: Pick a convenient test point (usually (0,0) if not on the line).
    • If the point satisfies the inequality, shade that side.
    • If not, shade the opposite side.

Quick Examples

  • y < x + 1
    1. Draw dashed line y = x + 1.
    2. Test (0,0):\; 0 < 1 → true → shade region containing (0,0).
  • 2x + 5 \le -2
    1. 2x \le -7
    2. x \le -\dfrac{7}{2}
  • Compound "or": x < -4 \;\text{or}\; x > 6 → two rays extending left from -4 and right from 6 (open circles).