In this lesson, two linear inequalities in two variables represent the constraints in a situation. Each pair of inequalities forms a system of inequalities.

A solution to a system of inequalities is any $(x,y)$ pair that makes both inequalities true, or any pair of values that simultaneously meet both constraints in the situation. The solution to the system is often best represented by a region on a graph.

Suppose there are two numbers, $x$ and $y$, and there are two things we know about them.

• The value of one number is more than double the value of the other.
• The sum of the two numbers is less than 10.

There are many possible pairs of numbers that meet the first constraint, for example: 1 and 3, or 4 and 9.

The same can be said about the second constraint, for example: 1 and 3, or 2.4 and 7.5.

The pair $x=1$ and $y=3$ meets both constraints, so it is a solution to the system.

The pair $x=4$ and $y=9$ meets the first constraint but not the second ($9 >2(4)$ is a true statement, but $4+9<10$ is not true.)

Remember that graphing is a great way to show all the possible solutions to an inequality, so let’s graph the solution region for each inequality.​​​​​​

A graph of an inequality on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. Dashed line starts below x axis and right of y axis, goes through negative 2 point 5 comma negative 5, 0 comma 0, and 2 point 5 comma 5. The region above the dashed line is shaded.

A graph of an inequality on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. A dashed line starts on the y axis at 10, goes through 5 comma 5, and ends on x axis at 10. The region below the dashed line is shaded.

Because we are looking for a pair of numbers that meet both constraints or make both inequalities true at the same time, we want to find points that are in the solution regions of both graphs.

To do that, we can graph both inequalities on the same coordinate plane.

The solution set to the system of inequalities is represented by the region where the two graphs overlap.

A graph of two intersecting inequalities on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. The first dashed line starts below x axis and right of y axis, goes through negative 2 point 5 comma negative 5, 0 comma 0, and 2 point 5 comma 5. The region above the dashed line is shaded. Second line at on the y axis at 10, goes through 5 comma 5, end on x axis at 10. The region below the dashed line is shaded.