Graphing an inequality and finding the feasible region

Graphing a linear inequality

• An inequality such as $2x + y \le 10$ describes a whole region of the plane, not a single line.
• Graphing it has two parts:
  • Draw the boundary line (the equation with $=$).
  • Shade the side that satisfies the inequality.

Drawing the boundary line

• Replace the inequality sign with $=$ to get the boundary: $2x + y = 10$.
• Plot it quickly using the intercepts:
  • $x$-intercept: set $y = 0$ → $x = 5$, giving $(5, 0)$.
  • $y$-intercept: set $x = 0$ → $y = 10$, giving $(0, 10)$.
• Use a solid line for $\le$ or $\ge$ (the boundary is included).
• Use a dashed line for $<$ or $>$ (the boundary is not included). NCEA constraints are almost always $\le$ or $\ge$, so lines are solid.

Deciding which side to shade

• Pick a test point not on the line — the origin $(0, 0)$ is easiest when the line does not pass through it.
• Substitute it into the inequality:
  • If the statement is true, shade the side containing the test point.
  • If it is false, shade the other side.
• For $2x + y \le 10$: at $(0,0)$, $0 \le 10$ is true, so shade the side with the origin.

Finding the feasible region

• The feasible region is the set of points that satisfy every constraint at once.
• Graph all the constraint lines on the same axes and shade each one.
• The feasible region is the area where all the shaded regions overlap.
• Tip: shade lightly, or shade the unwanted side out, so the surviving overlap stays clear.

Reading the vertices (corner points)

• The feasible region is a polygon (bounded) or an open shape (unbounded); its corners are the vertices.
• Each vertex is where two boundary lines cross — find it by solving those two equations simultaneously.
• The vertices matter because the optimum always sits at one of them (next page).