Chapter 5.6, Problem 2E

To determine

To compare: the graph of a linear inequality in two variables with the graph of a linear equation in two variables.

Explanation of Solution

Consider the linear inequality $x<y+3$ and the linear equation $x=y+3$ .

Steps to graph the linear inequality:

  $\begin{array}{l}x<y+3\\ y>x-3\end{array}$

Step 1: Graph $y=x-3$ . Use a dashed line because the inequality symbol is >.

Step 2: Test $\left(0,0\right)$

  $\begin{array}{l}y>x-3\\ 0>-3\end{array}$

Step 3: Because $\left(0,0\right)$ is a solution, shade the half plane that contains $\left(0,0\right)$ .

Graph of the linear inequality $x<y+3$

  

Graph of linear equation $y=x-3$ is shown below:

  

Comparison of the Graphs:

On the comparison of both graphs, there is a shaded area in the inequality graph which shows the solutions of the inequality lies in the shaded area.

Also, dashed lines is used which shows the points on the boundary line are not the solution of the inequality.