Chapter 5.6, Problem 27E

To determine

ToGraph:The inequality $-4x+y<-7$ in a coordinate plane.

Answer to Problem 27E

The graph of the given inequality is

  

Explanation of Solution

Given:

The inequality $-4x+y<-7$

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 $-4x+y<-7$

Consider the corresponding equation $-4x+y=-7$

Graphing the equation $-4x+y=-7$

For $x=0$

  $\begin{array}{c}-4x+y=-7\\ -4\left(0\right)+y=-7\text{ Substitute }x=0\\ 0+y=-7\\ y=-7\end{array}$

For $y=0$

  $\begin{array}{c}-4x+y=-7\\ -4x+\left(0\right)=-7\text{ Substitute }y=0\\ -4x=-7\\ \frac{-4x}{-4}=\frac{-7}{-4}\text{ Divide both sides by }-4\end{array}$

  $x=\frac{7}{4}$

The points on the equation $-4x+y=-7$ are

$x$	$0$	$\frac{7}{4}$
$y$	$-7$	$0$

Since, in the given inequality $-4x+y<-7$ , the symbol is $<$

So, plotting the graph of the equation $-4x+y=-7$ using a dashed line. We have,

  

Testing a point on either side of the dashed line

Consider a coordinate $\left(3,0\right)$ on right side of the dashed line.

Substituting the coordinate $\left(3,0\right)$ in the given inequality on both sides, we have

  $\begin{array}{c}-4x+y<-7\\ -4\left(3\right)+0<-7\text{ Substitute }\left(3,0\right)\\ -12<-7\text{ Which is TRUE}\end{array}$

Thus, $\left(3,0\right)$ is the solution of the given inequality.

Shading the half-plane that contains $\left(3,0\right)$ . The graph of the given inequality is

  

Conclusion:

The graph of the given inequality is