Chapter 5.6, Problem 30E

To determine

ToGraph:The inequality $-x+4y>-12$ in a coordinate plane.

Answer to Problem 30E

The graph of the given inequality is

  

Explanation of Solution

Given:

The inequality $-x+4y>-12$

Concept Used:

The graph of a linear inequality in two variables shows all the solutions of the inequality in a coordinate plane.

Calculation:

Given the inequality $-x+4y>-12$

Consider the corresponding equation $-x+4y=-12$

Graphing the equation $-x+4y=-12$

For $x=0$

  $\begin{array}{c}-x+4y=-12\\ -\left(0\right)+4y=-12\text{ Substitute }x=0\\ 4y=-12\\ \frac{4y}{4}=\frac{-12}{4}\text{ Dividing both sides by 4}\end{array}$

  $y=-3$

For $y=0$

  $\begin{array}{c}-x+4y=-12\\ -x+4\left(0\right)=-12\text{ Substitute }y=0\\ -x=-12\\ \left(-1\right)\left(-x\right)=\left(-1\right)\left(-12\right)\text{ Multiply both sides by }-1\end{array}$

  $x=12$

The points on the equation $-x+4y=-12$ are

$x$	$0$	$12$
$y$	$-3$	$0$

Since, in the given inequality $-x+4y>-12$ , the symbol is $>$

So, plotting the graph of the equation $-x+4y=-12$ using a dashed line. We have,

  

Testing a point on either side of the dashed line

Consider a coordinate $\left(0,0\right)$ on left side of the dashed line.

Substituting the coordinate $\left(0,0\right)$ in the given inequality on both sides, we have

  $\begin{array}{c}-x+4y>-12\\ -\left(0\right)+4\left(0\right)>-12\text{ Substitute }\left(0,0\right)\\ 0>-12\text{ Which is True}\end{array}$

Thus, $\left(0,0\right)$ is the solution of the given inequality.

Shading the half-plane that contains $\left(0,0\right)$ . The graph of the given inequality is

  

Conclusion:

The graph of the given inequality is