Chapter 5.7, Problem 15E

To determine

ToGraph:The system of linear inequalities $y\ge -5$ and $y-1<3x$

Answer to Problem 15E

The graph of the system of linear inequalities $y\ge -5$ and $y-1<3x$ is

  

Explanation of Solution

Given:

The system of linear inequalities $y\ge -5$ and $y-1<3x$

Concept Used:

The graph of asystem of linear inequalities in two variables shows all the solutions of the system.

Calculation:

Given the system of linear inequalities $y\ge -5$ and $y-1<3x$

Consider thelinear inequalities $y\ge -5$

The corresponding equation $y=-5$

Graphing the equation $y=-5$ ,

Plotting the graph of the equation $y=-5$ using a solid line. We have,

  

Testing a point on either side of the solid line

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

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

  $\begin{array}{c}y\ge -5\\ 0\ge -5\text{ Substitute }\left(0,0\right)\\ 0\ge -5\text{ Which is True}\end{array}$

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 inequality $y\ge -5$ is

  

Plotting the second inequality on the same above graph

Consider thesecond linear inequalities $y-1<3x$

The corresponding equation $y-1=3x$

For $x=0$

  $\begin{array}{c}y-1=3x\text{ Substitute }x=0\\ y-1=3\left(0\right)\\ y-1=0\\ y-1+1=0+1\text{ Add }1\text{ to both sides}\end{array}$

  $y=1$

For $y=0$

  $\begin{array}{c}y-1=3x\\ 0-1=3x\text{ Substitute }y=0\\ -1=3x\\ \frac{-1}{3}=\frac{3x}{3}\text{ Divide both sides by }3\end{array}$

  $x=\frac{-1}{3}$

The points on the equation $y-1=3x$ are

$x$	$0$	$\frac{-1}{3}$
$y$	$1$	$0$

Since, in the given inequality $y-1<3x$ , the symbol is $<$

So, plotting the graph of the equation $y-1=3x$ using a dashed line in the graph of the previous inequality. We have,

  

Testing a point on either side of the dashed line

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

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

  $\begin{array}{c}y-1<3x\\ 0-1<3\left(0\right)\text{ Substitute }\left(0,0\right)\\ -1<0\text{ Which is True}\end{array}$

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 inequality $y-1<3x$ is shown by red color

  

Plotting the intersection of the two shaded regions, we have

  

Conclusion:

The graph of the system of linear inequalities $y\ge -5$ and $y-1<3x$ is