Equations, Inequalities and Graphs (Copy)

4 Equations, Inequalities and Graphs – Cheat Sheet

4.1 Absolute Value (Modulus) Equations

• Definition: |x| = distance from 0; always ≥ 0
  • |x| = x if x ≥ 0
  • |x| = –x if x < 0

Types:

1. |ax + b| = c (c ≥ 0) →
   ax + b = c or ax + b = –c
   (If c < 0 → no solution)
2. |ax + b| = cx + d →
   ax + b = cx + d or ax + b = –(cx + d) → solve each.
3. |ax + b| = |cx + d| →
   ax + b = cx + d or ax + b = –(cx + d)
4. |ax² + bx + c| = d (d ≥ 0) →
   ax² + bx + c = d or ax² + bx + c = –d

4.2 Absolute Value Inequalities

1. k|ax + b| > c →
   ax + b > c/k or ax + b < –c/k
2. k|ax + b| ≤ c →
   –c/k ≤ ax + b ≤ c/k
3. k|ax + b| ≤ |cx + d| →
   Square both sides → solve resulting quadratic inequality.
4. |ax + b| ≤ cx + d →
   Check sign of (cx + d) and solve accordingly.
5. |ax² + bx + c| > d →
   ax² + bx + c > d or ax² + bx + c < –d
6. |ax² + bx + c| ≤ d →
   –d ≤ ax² + bx + c ≤ d

4.3 Substitution to Solve Related Equations

• Use substitution to reduce to quadratic form.
  Common substitutions:
  • u = ln(ax + b)
  • u = eˣ
  • u = x²
• Solve quadratic in u → back-substitute → solve for x.

4.4 Sketching Cubic Graphs and Moduli

• Cubic given as f(x) = (x – p)(x – q)(x – r)
  • Positive leading coefficient: starts ↓ (left), ends ↑ (right)
  • Negative leading coefficient: starts ↑ (left), ends ↓ (right)
• Steps:
  1. Find x-intercepts from factors.
  2. Find y-intercept (x = 0).
  3. Locate turning points (differentiate).
  4. For |f(x)|, reflect negative parts above x-axis.

4.5 Solving Cubic Inequalities

• Graphical method:
  1. Sketch y = f(x) and y = d.
  2. Find intersections → solve f(x) = d.
  3. Identify intervals where inequality holds.
• Forms: f(x) ≥ d, f(x) > d, f(x) ≤ d, f(x) < d

4 Equations, Inequalities and Graphs – Quick Reference Table