Chapter 11, Problem 90RE

To determine

Graph each system of inequalities.

Answer to Problem 90RE

Explanation of Solution

Given information:

Graph each system of inequalities.

  $\{\begin{array}{l}{y}^2\le x-1\\ x-y\le 3\end{array}$

Calculation:

Consider the sytem of inequalities.

  $\{\begin{array}{l}{y}^2\le x-1\\ x-y\le 3\end{array}$

Let take inequality,

  $y^2\le x-1$

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 first inequality of the given system,

  $\begin{array}{l}{\left(0\right)}^2\le 0-1\\ 0\le -1\end{array}$

Since $0\le -1$ is false statement point $\left(0,0\right)$ does not satisfying the inequality.

Hence the complete solution is set all the points which are opposite side of the curve as the point $\left(0,0\right)$ including the point on the curve $y^2\le x-1$ itself.

Now consider,

  $x-y\le 3$

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

  $\begin{array}{l}0-0\le 3\\ 0\le 3\end{array}$

Since $0\le 3$ is true statement point $\left(0,0\right)$ satisfying the inequality.

Hence the complete solution is set all the points which are on the same side of the line as the point $\left(0,0\right)$

So the solutions of all inequalities are satisfying and the graph is sketched in the rectangular Cartesian coordinate system as given below.

Hence, the graph of inequalities is as shown.