Chapter 5.6, Problem 26E

To determine

ToGraph:The inequality $y\le 3x-1$ in a coordinate plane.

Answer to Problem 26E

The graph of the given inequality is

  

Explanation of Solution

Given:

The inequality $y\le 3x-1$

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 $y\le 3x-1$

Consider the corresponding equation $y=3x-1$

Graphing the equation $y=3x-1$

For $x=0$

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

For $y=0$

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

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

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

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

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

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

  

Testing a point on either side of the dashed line

Consider a coordinate $\left(2,0\right)$ on right side of the solid line.

Substituting the coordinate $\left(2,0\right)$ in the given inequality on both sides, we have

  $\begin{array}{c}y\le 3x-1\\ 0\le 3\left(2\right)-1\text{ Substitute }\left(2,0\right)\\ 0\le 6-1\\ 0\le 5\text{ Which is TRUE}\end{array}$

Thus, $\left(2,0\right)$ is the solution of the given inequality.

Shading the half-plane that contains $\left(2,0\right)$ . The graph of the given inequality is

  

Conclusion:

The graph of the given inequality is