Chapter 5, Problem 1RE

Graph each inequality.

$x>2y-3$

To determine

To graph: The set of points that satisfy the inequality $x>2y-3$.

Explanation of Solution

Graph:

Consider the inequality $x>2y-3$.

First, sketch the line $x=2y-3$ by computing the ordered pairs as follows.

$\begin{array}{|cc|}\hline x& y\\ 0& 1.5\\ -3& 0\\ \hline\end{array}$

Hence, the line $x=2y-3$ passes through the points $\left(0,1.5\right)$ and $\left(-3,0\right)$.

Now, consider the inequality $x>2y-3$.

Check if the origin $\left(0,0\right)$ satisfies the inequality $x>2y-3$.

Substitute $0$ for $x$ and $y$ in the inequality $x>2y-3$.

$\begin{array}{l}x>2y-3\\ \left(0\right)\stackrel{?}{>}2\cdot \left(0\right)-3\\ 0\stackrel{?}{>}0-3\\ 0>-3\end{array}$

Since the point $\left(0,0\right)$ satisfies the inequality $x>2y-3$, the shaded region will be towards the origin or below the line $x=2y-3$.

Follow the steps given below to graph the inequality $x>2y-3$.

Step $1$ : Plot the points $\left(0,1.5\right)$ and $\left(-3,0\right)$.

Step $2$ : Sketch a dotted line $x=2y-3$ passing through the points $\left(0,1.5\right)$ and $\left(-3,0\right)$ as the inequality does not include the equality sign.

Step $3$ : Shade the region below the line $x=2y-3$.

The graph of the points that satisfy the inequality $x>2y-3$ is given below.

Hence, the solution set of the inequality is the region lying below the line $x=2y-3$.