Chapter 6.8, Problem 29PPS

To determine

To calculate:

Given system of inequality:

  $\begin{array}{l}2x-y\le -11\\ 3x-y\ge 12\end{array}$

Answer to Problem 29PPS

Given system of inequalities has no solution.

Explanation of Solution

Given information:

  $\begin{array}{l}2x-y\le -11\\ 3x-y\ge 12\end{array}$

Calculation:

Graph for the given inequality is as below:

  

Given system of inequality is,

  $\begin{array}{l}2x-y\le -11\\ 3x-y\ge 12\end{array}$

So isolate $y$ for $2x-y\le -11$ :

Subtract $2x$ from both sides,

  $2x-y-2x\le -11-2x$

After simplifying,

  $-y\le -11-2x$

Multiplies both sides by $-1$

  $\begin{array}{l}\left(-y\right)\left(-1\right)>-11\left(-1\right)-2x\left(-1\right)\\ =y\ge 11+2x\end{array}$

Now isolate $y$ for $3x-y\ge 12$ :

Subtract $3x$ from both sides,

  $3x-y-3x\ge 12-3x$

After simplifying,

  $-y\ge 12-3x$

Multiplied both sides by $-1$ ,

  $\left(-y\right)\left(-1\right)\le 12\left(-1\right)-3x\left(-1\right)$

After simplifying,

  $\begin{array}{l}y\le -12+3x\\ =\left[\begin{array}{l}y\ge 11+2x\\ y\le -12+3x\end{array}\right]\end{array}$

The above graph of $2x-y\le -11$ and $3x-y\ge 12$ are parallel lines.

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

Thus, the given system of inequalities has no solution.