Example 4.10.8

Graph \(y\gt2x+1\text{.}\)

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

  1. Graph the line \(y=2x+1\text{.}\) Because the inequality symbol is \(\gt\) (instead of \(\ge\)), the line should be dashed (instead of solid).
  2. Next, we need to decide whether to shade the region above \(y=2x+1\) or below it. We will choose a point to test whether \(y\gt2x+1\) is true. As long as the line doesn't cross \((0,0)\text{,}\) we will use \((0,0)\) to test because the number \(0\) is the easiest number for calculation.
    \begin{align*} y\amp\gt2x+1\\ 0\amp\stackrel{?}{\gt}2(0)+1\\ 0\amp\stackrel{\text{no}}{\gt}1 \end{align*}
    Because \(0\gt1\) is not true, the point \((0,0)\) is not a solution and should not be shaded. As a result, we shade the region without \((0,0)\text{.}\)
Figure 4.10.9 Step 1 of graphing \(y\gt2x+1\)
Figure 4.10.10 Complete graph of \(y\gt2x+1\)