Quadratic Inequalities and Solving Inequalities

Quadratic Inequalities

• A quadratic inequality is an inequality of the form ax^2 + bx + c > 0, ax^2 + bx + c ≥ 0, ax^2 + bx + c < 0, or ax^2 + bx + c ≤ 0, where a, b, and c are real numbers and a ≠ 0.
• The solutions to a quadratic inequality can be found by:
  • Factoring the left-hand side, if possible
  • Using the Quadratic Formula to find the roots of the related quadratic equation
  • Graphing the related quadratic function on a number line

Solving Inequalities

• Solving an inequality involves finding the values of the variable that make the inequality true.
• The steps to solve an inequality are:
  1. Simplify the inequality by combining like terms and eliminating any parentheses or other grouping symbols.
  2. Add or subtract the same value to both sides of the inequality to isolate the variable term.
  3. Multiply or divide both sides of the inequality by a coefficient of the variable, if necessary, to solve for the variable.
     • Note: When multiplying or dividing both sides by a negative number, the direction of the inequality symbol must be flipped.
  4. Check the solution by plugging it back into the original inequality.

Properties of Inequalities

• The following properties can be used to solve inequalities:
  • Addition Property: If a > b, then a + c > b + c
  • Subtraction Property: If a > b, then a - c > b - c
  • Multiplication Property: If a > b and c > 0, then ac > bc
    • If a > b and c < 0, then ac < bc
  • Transitive Property: If a > b and b > c, then a > c

Graphing Inequalities

• Inequalities can be graphed on a number line to visualize the solution.
• Open circles or parentheses are used to indicate strict inequalities (e.g., < or >), while closed circles or brackets are used to indicate inclusive inequalities (e.g., ≤ or ≥).
• The direction of the inequality symbol indicates the direction of the solution on the number line.