Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 9: Introduction to Inequalities

Lesson 4: Graphing Inequalities

In this Grade 4 AoPS Introduction to Algebra lesson, students learn how to graph two-variable linear inequalities on a coordinate plane by plotting a boundary line and shading the solution region. The lesson covers key techniques such as using a test point like the origin to determine which side to shade, distinguishing between solid and dashed boundary lines for nonstrict versus strict inequalities, and finding the intersection region that satisfies two simultaneous linear inequalities. This chapter from the Introduction to Inequalities unit builds the foundational graphing skills needed for AMC 8 and AMC 10 competition problems.

Section 1

The Boundary Line

Property

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

\begin{array}{ll} Ax + By < C & Ax + By \leq C \\ Ax + By > C & Ax + By \geq C \\ \text{Boundary line is } Ax + By = C & \text{Boundary line is } Ax + By = C \\ \text{Boundary line is not included in solution.} & \text{Boundary line is included in solution.} \\ \text{Boundary line is dashed.} & \text{Boundary line is solid.} \end{array}

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.

Section 2

Graphing Linear Inequalities

Property

A linear inequality can be written in the form

ax + by + c \leq 0 \quad \text{or} \quad ax + by + c \geq 0

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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

The Boundary Line

Property

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

\begin{array}{ll} Ax + By < C & Ax + By \leq C \\ Ax + By > C & Ax + By \geq C \\ \text{Boundary line is } Ax + By = C & \text{Boundary line is } Ax + By = C \\ \text{Boundary line is not included in solution.} & \text{Boundary line is included in solution.} \\ \text{Boundary line is dashed.} & \text{Boundary line is solid.} \end{array}

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.

Section 2

Graphing Linear Inequalities

Property

A linear inequality can be written in the form

ax + by + c \leq 0 \quad \text{or} \quad ax + by + c \geq 0

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.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

The Boundary Line

Property

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

\begin{array}{ll} Ax + By < C & Ax + By \leq C \\ Ax + By > C & Ax + By \geq C \\ \text{Boundary line is } Ax + By = C & \text{Boundary line is } Ax + By = C \\ \text{Boundary line is not included in solution.} & \text{Boundary line is included in solution.} \\ \text{Boundary line is dashed.} & \text{Boundary line is solid.} \end{array}

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.

Section 2

Graphing Linear Inequalities

Property

A linear inequality can be written in the form

ax + by + c \leq 0 \quad \text{or} \quad ax + by + c \geq 0

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.