Example: Graphing an Inequality for a Parabola

Graph the inequality [latex]y>{x}^{2}+1[/latex].

Answer: First, graph the corresponding equation [latex]y={x}^{2}+1[/latex]. Since [latex]y>{x}^{2}+1[/latex] has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Let’s test the points [latex]\left(0,2\right)[/latex] and [latex]\left(2,0\right)[/latex]. One point is clearly inside the parabola and the other point is clearly outside.

[latex]\begin{array}{ll}y>{x}^{2}+1\hfill & \hfill \\ 2>{\left(0\right)}^{2}+1\hfill & \hfill \\ 2>1\hfill & \text{True}\hfill \\ \hfill & \hfill \\ \hfill & \hfill \\ \hfill & \hfill \\ 0>{\left(2\right)}^{2}+1\hfill & \hfill \\ 0>5\hfill & \text{False}\hfill \end{array}[/latex]

The graph is shown below. We can see that the solution set consists of all points inside the parabola, but not on the graph itself.

Example: Graphing a System of Inequalities

Graph the given system of inequalities.

[latex]\begin{array}{r}\hfill {x}^{2}-y\le 0\\ \hfill 2{x}^{2}+y\le 12\end{array}[/latex]

Answer: These two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable [latex]y[/latex] will be eliminated. Then we solve for [latex]x[/latex].

[latex]\begin{array}\hfill x^{2}−y=0 \\ 2x^{2}+y=12 \\ \text{___________} \\ 3x^{2}=12 \\ x^{2}=4 \\ x=\pm 2\end{array}[/latex]

Substitute the x-values into one of the equations and solve for [latex]y[/latex].

[latex]\begin{array}{r}\hfill {x}^{2}-y=0\\ \hfill {\left(2\right)}^{2}-y=0\\ \hfill 4-y=0\\ \hfill y=4\\ \hfill \\ \hfill {\left(-2\right)}^{2}-y=0\\ \hfill 4-y=0\\ \hfill y=4\end{array}[/latex]

The two points of intersection are [latex]\left(2,4\right)[/latex] and [latex]\left(-2,4\right)[/latex]. Notice that the equations can be rewritten as follows.

[latex]\begin{array}{l}{x}^{2}-y\le 0\hfill \\ {x}^{2}\le y\hfill \\ y\ge {x}^{2}\hfill \\ \hfill \\ 2{x}^{2}+y\le 12\hfill \\ y\le -2{x}^{2}+12\hfill \end{array}[/latex]

Graph each inequality. The feasible region is the region between the two equations bounded by [latex]2{x}^{2}+y\le 12[/latex] on the top and [latex]{x}^{2}-y\le 0[/latex] on the bottom.