Chapter 7, Problem 1RE

To determine

To graph: Graph of the given linear inequality: $y\le 3x+2$

Explanation of Solution

Given:

$y\le 3x+2$

Graph:

The provided linear inequality is

$y\le 3x+2$

Replace inequality symbol with equality and rewrite the equation as follows:

$y=3x+2$

To find x-intercept, substitute $y=0$ as follows:

$\begin{array}{c}y=3x+2\\ 0=3x+2\\ 3x=-2\\ x=\frac{-2}{3}\end{array}$

To find y-intercept, substitute $x=0$ as follows:

$\begin{array}{c}y=3x+2\\ y=\left(3\times 0\right)+2\\ y=2\end{array}$

Now, take origin as test point and check which region to shade:

$\begin{array}{c}y\le 3x+2\\ 0\le \left(3\times 0\right)+2\\ 0\le 2\end{array}$

As the above expression is true, the shaded region will contain the test point, that is, origin. So, the line passes through points $\left(\frac{-2}{3},0\right)$ and $\left(0,2\right)$. Plot the graph as follows:

Hence, the graph of the provided inequality line passes through points $\left(\frac{-2}{3},0\right)$ and $\left(0,2\right)$. Also, the shaded region does contain origin.