Chapter 6.8, Problem 28PPS

To determine

To calculate:

Given system of inequality:

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

Answer to Problem 28PPS

Given system of inequalities has no solution.

Explanation of Solution

Given information:

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

Calculation:

Graph for the given inequality is as below:

  

Given system of inequality is,

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

So isolate $y$ for $3x-y<-2$ :

Subtract $3x$ from both sides,

  $3x-y-3x<-2-3x$

After simplifying,

  $-y<-2-3x$

Multiplies both sides by $-1$

  $\begin{array}{l}\left(-y\right)\left(-1\right)>-2\left(-1\right)-3x\left(-1\right)\\ =y>2+3x\end{array}$

Now isolate $y$ for $3x-y<1$ :

Subtract $3x$ from both sides,

  $3x-y-3x<1-3x$

After simplifying,

  $-y<1-3x$

Multiplied both sides by $-1$ ,

  $\left(-y\right)\left(-1\right)>1\left(-1\right)-3x\left(-1\right)$

After simplifying,

  $\begin{array}{l}y>-1+3x\\ =\left[\begin{array}{l}y>2+3x\\ y>-1+3x\end{array}\right]\end{array}$

The above graph of $3x-y<-2$ and $3x-y<1$ are parallel lines.

So, the two regions do not intersect at any point.

Thus, the given system of inequalities has no solution.