4.7 Graphs of linear inequalities

Verify solutions to an inequality in two variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.

Do you remember that an inequality with one variable had many solutions? The solution to the inequality $x>3$ is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See [link] .

Similarly, inequalities in two variables have many solutions. Any ordered pair $\left(x,y\right)$ that makes the inequality true when we substitute in the values is a solution of the inequality.

Recognize the relation between the solutions of an inequality and its graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in [link] again. The point $x=3$ separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See [link] .

The solution to $x>3$ is the shaded part of the number line to the right of $x=3$ .

Similarly, the line $y=x+4$ separates the plane into two regions. On one side of the line are points with $y<x+4$ . On the other side of the line are the points with $y>x+4$ . We call the line $y=x+4$ a boundary line.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not $a$ is included in the solution: