Chapter 11, Problem 91RE

To determine

Graph each system of inequalities.

Answer to Problem 91RE

Explanation of Solution

Given information:

Graph each system of inequalities.

  $\{\begin{array}{l}y\le x^2\\ xy\le 4\end{array}$

Calculation:

Consider the sytem of inequalities.

  $\{\begin{array}{l}y\le x^2\\ xy\le 4\end{array}$

To graph the given system of inequalities, first graph parabola $y=x^2$ and then equation

  $xy\le 4$ .

To determine the part of the graph to be shaded use a test point $\left(1,2\right)$ .

Substitute $x=1,y=2$ in the first inequality of the given system,

  $\begin{array}{l}\left(2\right)\le 1^2\\ 2\le 1\\ but\\ 2>1\end{array}$

Since $2\le 1$ is false statement point $\left(1,2\right)$ does not satisfying the inequality. So we shade the region outside the parabola.

Now consider,

  $xy\le 4$

To determine the part of the graph to be shaded use a test point $\left(0,0\right)$ .

Substitute $x=0,y=0$ in the inequality of the given system,

  $\begin{array}{l}\left(0\right)\left(0\right)\le 4\\ 0\le 4\end{array}$

Since $0\le 4$ is true statement point $\left(0,0\right)$ satisfying the inequality.so we shade the region below the graph of $xy=4$ .

Now graph the system of inequalities.

Hence the intersection of shaded region represents the graph of inequalities.