Inequalities & Optimization

Cheat Sheet · Grade 11 Math
Emil Oliversen
SOLVING INEQUALITIES
KEY RULE: Flip sign when × or ÷ by negative!
Simple
3x+4 < 22 → x < 6
Negative div
−5x ≥ 5 → x ≤ −1 (flip!)
Double
9 ≤ x ≤ 13
GRAPHING (1 VARIABLE)
≤ or ≥
● closed dot
< or >
○ open dot
GRAPHING (2 VARIABLES)
  • ≤ or ≥ → solid boundary line
  • < or > → dashed boundary line
  • Test point (0,0): satisfies? shade that side
  • Does NOT satisfy? shade other side
SYSTEMS OF INEQUALITIES
  • Graph each inequality separately
  • The overlapping region = solution set
  • This region = polygon of constraints
  • Vertices of polygon are key for optimization
LINEAR PROGRAMMING
Optimal value always occurs at a vertex of the polygon of constraints
OPTIMIZATION STEPS
  • Define variables x and y
  • Write objective function (e.g. Max = 20x+30y)
  • Write constraint inequalities
  • Graph → find polygon of constraints
  • Find ALL vertex coordinates
  • Substitute each vertex into objective
  • State max/min and what it means
OPEN vs CLOSED VERTICES
A vertex on a dashed line is NOT included → cannot be optimal
MULTIPLE SOLUTIONS
If two adjacent vertices give same value, ALL points on the edge between them are also optimal
WORKED EXAMPLE
Salon: $20/haircut (x), $30/colour (y)
Max = 20x + 30y
A(3,2)→$120 B(3,8)→$300 C(8,3)→$250
Max = $300 at B(3,8)
COMMON MISTAKES
  • Forget to flip sign ÷ by negative
  • Wrong line type (solid vs dashed)
  • Test point wrong side → wrong shade
  • Miss a vertex → wrong optimum
  • Only check some vertices
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