Chapter 6.8, Problem 22PPS

To determine

To calculate:

Given system of inequality:

  $\begin{array}{l}5x-y<-6\\ 3x-y\ge 4\end{array}$

Answer to Problem 22PPS

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

  

Explanation of Solution

Given information:

  $\begin{array}{l}5x-y<-6\\ 3x-y\ge 4\end{array}$

Calculation:

Graph for the given inequality is as below:

  

Given system of inequality is,

  $\begin{array}{l}5x-y<-6\\ 3x-y\ge 4\end{array}$

So isolate $y$ for $5x-y<-6$ :

Subtract $5x$ from both sides,

  $5x-y-5x<-6-5x$

Multiply both sides by $-1$

  $\left(-y\right)\left(-1\right)>-6\left(-1\right)-5x\left(-1\right)$

After simplifying,

  $\begin{array}{l}y>6+5x\\ =\left[\begin{array}{l}y>6+5x\\ y\le -4+3x\end{array}\right]\end{array}$

Now isolate $y$ for $3x-y\ge 4$ :

Subtract $3x$ from both sides,

  $\begin{array}{l}3x-y-3x\ge 4-3x\\ =-y\ge 4-3x\end{array}$

Multiply both sides by $-1$

  $\left(-y\right)\left(-1\right)\le 4\left(-1\right)-3x\left(-1\right)$

After simplifying,

  $\begin{array}{l}y\le -4+3x\\ =\left[\begin{array}{l}y>6+5x\\ y\le -4+3x\end{array}\right]\end{array}$

The above graph of $5x-y<-6$ and $3x-y\ge 4$ are not parallel lines.

So, the two regions intersect at any point.

Intersection point of the given inequality is $\left(-5,-19\right)$

Thus, the given system of inequalities has a solution.