Lesson 4.7: Graphs of Linear Inequalities

New Concept

This lesson explores linear inequalities in two variables. We'll connect solutions to their graphical representation by identifying the boundary line and shading the correct half-plane to represent all possible solutions.

What’s next

Next, you will verify solutions with interactive examples. Then, you'll master graphing inequalities step-by-step using practice cards and challenge problems.

Property

A linear inequality is an inequality that can be written in one of the following forms:
$A x + B y > C$
$A x + B y \geq C$
$A x + B y < C$
$A x + B y \leq C$
where $A$ and $B$ are not both zero.

Examples

• The statement $3x - y < 7$ is a linear inequality.
• The statement $y \geq 2x + 5$ is a linear inequality.
• The statement $x < 4$ is a linear inequality where the coefficient of $y$ is $0$.

Explanation

A linear inequality describes a relationship that isn't strictly equal. Instead of a single line of solutions, it represents a whole region on the coordinate plane. Think of it as a 'more than' or 'less than' zone with infinite solutions.

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

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

For $Ax + By < C$ or $Ax + By > C$, the boundary line is not included in the solution, and the line is dashed.

For $Ax + By \leq C$ or $Ax + By \geq C$, the boundary line is included in the solution, and the line is solid.

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.

Property

Inequalities with only one variable have boundary lines that are horizontal or vertical.
An inequality with only $y$ (e.g., $y>3$) has a horizontal boundary line.
An inequality with only $x$ (e.g., $x \leq 5$) has a vertical boundary line.

Examples

• To graph the inequality $y < 4$: The boundary is a dashed horizontal line at $y=4$. Since the test point $(0,0)$ makes $0 < 4$ true, we shade the region below the line.

• To graph the inequality $x \geq -2$: The boundary is a solid vertical line at $x=-2$. Since the test point $(0,0)$ makes $0 \geq -2$ true, we shade the region to the right of the line.