Verifying Solutions to An Inequality in Two Variables
Verifying 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.
Linear Inequality
A linear inequality is an inequality that can be written in one of the following forms:
where \(A\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}B\) are not both zero.
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 the figure below.
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.
Solution of a Linear Inequality
An ordered pair \(\left(x,y\right)\) is a solution of a linear inequality if the inequality is true when we substitute the values of x and y.
Example
Determine whether each ordered pair is a solution to the inequality \(y>x+4\):
\(\left(0,0\right)\) \(\left(1,6\right)\) \(\left(2,6\right)\) \(\left(-5,-15\right)\) \(\left(-8,12\right)\)
Solution
-
\(\left(0,0\right)\) Simplify.
So, \(\left(0,0\right)\) is not a solution to \(y>x+4\). -
\(\left(1,6\right)\) Simplify.
So, \(\left(1,6\right)\) is a solution to \(y>x+4\). -
\(\left(2,6\right)\) Simplify.
So, \(\left(2,6\right)\) is not a solution to \(y>x+4\). -
\(\left(-5,-15\right)\) Simplify.
So, \(\left(-5,-15\right)\) is not a solution to \(y>x+4\). -
\(\left(-8,12\right)\) Simplify.
So, \(\left(-8,12\right)\) is a solution to \(y>x+4\).
This lesson is part of:
Graphs and Equations