Chapter 2.8, Problem 40PPSE

Solution

To find: To write an inequality for graph given.

  $y\le 2x+2$

Given information: To write an equality for the graph given below:

  

Explanation:

From the graph, the intercepts are $\left(-1,0\right)$ and $\left(0,2\right)$ :

Use slope formula to find the slope $\left(m\right)$ :

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\left[\left(x_1,y_1\right)=\left(-1,0\right),\left(x_2,y_2\right)=\left(0,2\right)\right]\\ \Rightarrow =\frac{2-0}{0-\left(-1\right)}\\ \Rightarrow =2\end{array}$

Substitute $\left(x_1,y_1\right)=\left(-1,0\right)$ and $m=2$ into the point-slope form to get equation of boundary line:

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\\ y-0=2\left(x-\left(-1\right)\right)\\ \Rightarrow y=2x+2\end{array}$

The boundary line is the graph of $y=2x+2$

Since, the solution is shaded below the boundary line, so the inequality is $<$ or $\le$ .

Since the boundary line is solid line, the correct inequality is $\le$ :

Therefore, the inequality for the graph is $y\le 2x+2$.