AP Precalculus: Nonlinear Inequalities Formulas & Methods

1. Graph Solutions to Quadratic Inequalities

• Rewrite in standard form: \( ax^2 + bx + c \; {\color{#a33c61}\gtrless} \; 0 \)
• Find roots/zeros: set \( ax^2 + bx + c = 0 \), solve for \( x \)
• Mark roots on number line, test sign in each region (between and outside of roots)
• Shaded regions below x-axis: \( <0 \); above x-axis: \( >0 \)
• Solution intervals: where inequality is true

2. Solve Quadratic Inequalities

• Steps:
  1. Write inequality: \( ax^2+bx+c\, \gtrless 0 \)
  2. Set corresponding equation to zero and factor/solve for \( x \)
  3. Find critical points (roots)
  4. Test sign of the expression in intervals between/around roots
  5. Write solution in interval notation
• Example: \( x^2 - 5x + 6 < 0 \)
  Roots at \( x = 2, 3 \).
  Test intervals \( (-\infty, 2), (2,3), (3, \infty) \)—solution: \( (2,3) \)

3. Graph Solutions to Higher-Degree Inequalities

• Write as \( f(x) \gtrless 0 \) for polynomial \( f(x) \)
• Find all real roots (x-intercepts)
• Divide number line at roots
• Test value in each interval to determine where \( f(x) \) is + or -
• Shade intervals where inequality holds

4. Solve Higher-Degree Inequalities

• Factor \( f(x) \) completely if possible
• Identify multiplicity:
  • Even multiplicity: graph touches x-axis, sign does NOT change
  • Odd multiplicity: graph crosses, sign does change
• Follow same sign-testing process as quadratic
• Express complete solution in interval notation