Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.

Linear 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 made a profit.

Recall that an inequality with one variable had many solutions. For example, 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 the figure below.

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

Example

Determine whether each ordered pair is a solution to the inequality $y>x+4$:

1. $(0,0)$
2. $(1,6)$
3. $(2,6)$
4. $(-5,-15)$
5. $(-8,12)$

Solution

1. No; Substitute 0 for $x$ and 0 for $y$.

$y>x+4$

$0>0+4$

$0\ngtr 4$

2. Yes; Substitute 1 for $x$ and 6 for $y$.

$y>x+4$

$6>1+4$

$6>5$

3. No; Substitute 2 for $x$ and 6 for $y$.

$y>x+4$

$6>2+4$

$6\ngtr 6$

4. No; Substitute -5 for $x$ and -15 for $y$.

$y>x+4$

$-15>-5+4$

$-15\ngtr 6-1$

5. Yes; Substitute -8 for $x$ and 12 for $y$.

$y>x+4$

$12>-8+4$

$12>-4$