Lesson 3.4: Graph Linear Inequalities in Two Variables

New Concept

We're expanding from single-variable inequalities to those with two variables, like $Ax + By > C$. This lesson teaches you how to graph these inequalities, where the solution isn't just a line, but a whole region on the coordinate plane.

What’s next

Next up, you'll tackle interactive graphing examples, practice identifying solution regions, and solve real-world problems. Let's start graphing!

Property

A linear inequality is an inequality that can be written in one of the following forms:

Where $A$ and $B$ are not both zero.

Examples

• An ordered pair $(x, y)$ is a solution if it makes the inequality true. For $y > x + 4$, the point $(1, 6)$ is a solution because $6 > 1 + 4$ is true.

• For the same inequality $y > x + 4$, the point $(2, 6)$ is not a solution because $6 > 2 + 4$ simplifies to $6 > 6$, which is false.

Property

An ordered pair $(x, y)$ is a solution to a linear inequality if the inequality is true when we substitute the values of $x$ and $y$.

Examples

• Is $(2, 5)$ a solution to $y < 2x + 3$? Substitute the values: $5 \stackrel{?}{<} 2(2) + 3$. This simplifies to $5 < 7$, which is true. So, $(2, 5)$ is a solution.

• Is $(-1, 4)$ a solution to $y < 2x + 3$? Substitute the values: $4 \stackrel{?}{<} 2(-1) + 3$. This simplifies to $4 < 1$, which is false. So, $(-1, 4)$ is not a solution.

Property

The line with equation $Ax + By = C$ is the boundary line that separates the region where $Ax + By > C$ from the region where $Ax + By < C$.

Examples

• For the inequality $y \leq 3x - 1$, the boundary line is $y = 3x - 1$. Since the inequality is 'less than or equal to' ($\leq$), the line is solid.

• For the inequality $x + 5y > 10$, the boundary line is $x + 5y = 10$. Since the inequality is 'greater than' ($>$), the line is dashed.

Property

How to graph a linear inequality in two variables.

• Step 1. Identify and graph the boundary line. If the inequality is $\leq$ or $\geq$, the boundary line is solid. If the inequality is $<$ or $>$, the boundary line is dashed.
• Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
• Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side.

Examples

• To graph $y \geq 2x - 3$: Draw the solid line $y = 2x - 3$. Test $(0,0)$: $0 \geq 2(0) - 3$ is $0 \geq -3$, which is true. Shade the side containing $(0,0)$.

• To graph $x - 3y < 6$: Draw the dashed line $x - 3y = 6$. Test $(0,0)$: $0 - 3(0) < 6$ is $0 < 6$, which is true. Shade the side containing $(0,0)$.