Lesson 6: Graphing Linear Inequalities in Two Variables

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 of a linear inequality if the inequality is true when we substitute the values of $x$ and $y$.

Examples

• To check if $(1, 4)$ is a solution to $y > 3x - 1$, substitute: $4 > 3(1) - 1$ becomes $4 > 2$. This is true, so $(1, 4)$ is a solution.
• To check if $(2, -1)$ is a solution to $2x + y \leq 3$, substitute: $2(2) + (-1) \leq 3$ becomes $3 \leq 3$. This is true, so $(2, -1)$ is a solution.
• To check if $(0, 0)$ is a solution to $x - 5y > 1$, substitute: $0 - 5(0) > 1$ becomes $0 > 1$. This is false, so $(0, 0)$ is not a solution.

Explanation

A solution is any point $(x, y)$ that makes the inequality true. Unlike equations that have solutions on a line, inequalities have solutions in a whole shaded region. Any point in that region works!

Property

To graph a linear inequality, follow these steps:

1. Identify and graph the boundary line. Use a solid line for $\leq$ or $\geq$, and a dashed line for $<$ or $>$.
2. Test a point that is not on the boundary line to see if it is a solution.
3. If the test point is a solution, shade the side of the line that includes it. If not, shade the opposite side.

Examples

• To graph $y > 2x - 1$: Draw a dashed line for $y = 2x - 1$. Test $(0,0)$: $0 > -1$ is true. Shade the side containing $(0,0)$.
• To graph $x + 3y \leq 6$: Draw a solid line for $x + 3y = 6$. Test $(0,0)$: $0 \leq 6$ is true. Shade the side containing $(0,0)$.
• To graph $x < -2$: Draw a dashed vertical line at $x = -2$. Test $(0,0)$: $0 < -2$ is false. Shade the side that does not contain $(0,0)$, which is the left side.

Explanation

Graphing an inequality is a three-step process: draw the boundary line (solid or dashed), pick a test point (like $(0,0)$) to see which side is true, and then shade that entire region to show all possible solutions.