What affects the inequality sign when rearranging a quadratic inequality?

• The inequality sign is?unchanged?by...
  • Adding/subtracting?a term to both sides
  • Multiplying/dividing?both sides by a?positive term
• The inequality sign?flips?(< changes to >) when...
  • Multiplying/dividing?both sides by a?negative term

How do I solve a quadratic inequality?

• STEP 1:?Rearrange?the inequality into quadratic form with a?positive squared term
  • ax²?+?bx?+?c > 0
  • ax²?+?bx?+?c ≥ 0
  • ax²?+?bx?+?c < 0
  • ax²?+?bx?+?c ≤ 0
• STEP 2: Find the?roots?of the quadratic equation
  • Solve?ax²?+?bx?+?c?= 0 to get?x₁?and?x₂?where?x₁?<?x₂
• STEP 3:?Sketch?a graph of the quadratic and label the roots
  • As the squared term is positive it will be?concave up?so "U" shaped
• STEP 4:?Identify?the?region?that satisfies the inequality
  • If you want the graph to be?above the?x-axis?then choose the region to be the?two intervals outside?of the two roots
  • If you want the graph to be?below the?x-axis?then choose the region to be the?interval between?the two roots
  • For?ax²?+?bx?+?c > 0
    • The solution is?x?<?x₁?or?x?>?x₂
  • For?ax²?+?bx?+?c ≥ 0
    • The solution is?x?≤?x₁?or?x?≥?x₂
  • For?ax²?+?bx?+?c < 0
    • The solution is?x_1?<?x?<?x-₂
  • For?ax²?+?bx?+?c ≤ 0
    • The solution is?x_1?≤?x?≤?x-₂

How do I solve a quadratic inequality of the form (x -?h)²?<?n?or (x -?h)²?>?n?