Chapter 6.8, Problem 23PPS

To determine

To calculate:

Given system of inequality:

  $\begin{array}{l}x-y\le 8\\ y<3x\end{array}$

Answer to Problem 23PPS

Solution of the function will be area under shaded region shown by graph:

  

Explanation of Solution

Given information:

  $\begin{array}{l}x-y\le 8\\ y<3x\end{array}$

Calculation:

Graph for the given inequality is as below:

  

Given system of inequality is,

  $\begin{array}{l}x-y\le 8\\ y<3x\end{array}$

So isolate $y$ for $x-y\le 8$ :

Subtract $x$ from both sides,

  $\begin{array}{l}x-y-x\le 8-x\\ =-y\le 8-x\end{array}$

Multiply both sides by $-1$

  $\left(-y\right)\left(-1\right)\ge 8\left(-1\right)-x\left(-1\right)$

After simplifying,

  $\begin{array}{l}y>-8+x\\ =\left[\begin{array}{l}y>-8+x\\ y<3x\end{array}\right]\end{array}$

The above graph of $x-y\le 8$ and $y<3x$ are not parallel lines.

So, the two regions intersect at any point.

Intersection point of the given inequality is $\left(-4,-12\right)$

Thus, the given system of inequalities has a solution.