Example4.10.11

Graph \(y\leq -\frac{5}{3}x+2\text{.}\)

There are two steps to graphing this linear inequality in two variables.

  1. Graph the line \(y= -\frac{5}{3}x+2\text{.}\) Because the inequality symbol is \(\leq\) (instead of \(\lt\)), the line should be solid.
  2. Next, we need to decide whether to shade the region above \(y= -\frac{5}{3}x+2\) or below it. We will choose a point to test whether \(y\leq -\frac{5}{3}x+2\) is true there. Using \((0,0)\) as a test point:
    \begin{align*} y\amp\leq -\frac{5}{3}x+2\\ 0\amp\stackrel{?}{\leq}-\frac{5}{3}(0)+2\\ 0\amp\stackrel{\checkmark}{\leq}2 \end{align*}
    Because \(0\leq2\) is true, the point \((0,0)\) is a solution. As a result, we shade the region with \((0,0)\text{.}\)
A coordinate plane with y=-5/3x+2 graphed as a solid line.
A coordinate plane with y=-5/3x+2 graphed as a solid line; the region below the line is shaded.
Figure4.10.12Step 1 of graphing \(y\leq -\frac{5}{3}x+2\)
Figure4.10.13Complete graph of \(y\leq -\frac{5}{3}x+2\)
in-context