Lesson 7: Systems of Linear Inequalities

Property

Two or more linear inequalities grouped together form a system of linear inequalities.

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities.

Examples

Determine whether the ordered pair is a solution to the system:

• Is $(2, 3)$ a solution? For the first inequality, $2 + 2(3) \leq 8$, which is $8 \leq 8$ (True). For the second, $3(2) - 3 > 5$, which is $3 > 5$ (False). Since one is false, $(2, 3)$ is not a solution.
• Is $(4, 1)$ a solution? For the first inequality, $4 + 2(1) \leq 8$, which is $6 \leq 8$ (True). For the second, $3(4) - 1 > 5$, which is $11 > 5$ (True). Since both are true, $(4, 1)$ is a solution.
• Is $(-1, -10)$ a solution? For the first inequality, $-1 + 2(-10) \leq 8$, which is $-21 \leq 8$ (True). For the second, $3(-1) - (-10) > 5$, which is $7 > 5$ (True). Since both are true, $(-1, -10)$ is a solution.

Property

To check if an ordered pair $(x, y)$ is a solution to a system of linear inequalities, substitute the $x$ and $y$ values into each inequality in the system. The ordered pair is a solution if and only if it satisfies all inequalities in the system.

Examples

Property

A linear inequality can be written in the form

The solutions consist of a boundary line and a half-plane. If the inequality is equivalent to $y \geq mx + b$, shade the half-plane above the line. If it is equivalent to $y \leq mx + b$, shade below the line. An inequality with $>$ or $<$ is strict, and its boundary line is dashed.

Examples

• To graph $3x - y > 6$, we first solve for $y$. This gives $-y > -3x + 6$, which becomes $y < 3x - 6$ after dividing by $-1$. We draw a dashed line for $y = 3x - 6$ and shade the half-plane below it.

• To graph $x + 2y \leq 8$, we solve for $y$ to get $2y \leq -x + 8$, or $y \leq -\frac{1}{2}x + 4$. We draw a solid line for $y = -\frac{1}{2}x + 4$ and shade the region below it.