Chapter 7, Problem 78RE

To determine

To sketch: the graph of the solution set of the system of inequalities.

Explanation of Solution

Given information:

Given system is

  $\{\begin{array}{l}y\le 6-2x-x^2\\ y\ge x+6\end{array}$

Calculation:

Let us consider the following inequalities

  $\{\begin{array}{l}y\le 6-2x-x^2\\ y\ge x+6\end{array}$

Consider the inequalities

  $\begin{array}{l}y\le 6-2x-x^2\mathrm{...............}(1)\\ y\ge x+\mathrm{6....................}(2)\end{array}$

Find the point of intersection of the above two equations

Equate the $y$ value from equation(1) and (2) to get

  $\begin{array}{c}x+\cancel{6}=\cancel{6}-2x-x^2\\ x^2+3x=0\\ x\left(x+3\right)=0\\ x=0,-3\end{array}$

Put $x=0$ in the equation (2) to get

  $\begin{array}{c}y=x+6\\ =0+6\\ =6\end{array}$

Put $x=-3$ in the equation (2) to get

  $\begin{array}{c}y=x+6\\ =-2+6\\ =4\end{array}$

Hence, the points of intersections are $\left(0,0\right)$ and $\left(-3,4\right)$

The graph and the vertices of the solution set of the above inequalities are shown below