Graph the Inequality Calculator

Input your linear inequality and we'll provide a textual representation of its solution set and how to graph it.

Understanding Linear Inequalities and Their Graphs

Linear inequalities are fundamental concepts in algebra and are widely used to model real-world situations where relationships are not exact but rather defined by ranges or boundaries. Unlike linear equations, which represent a single line on a graph, linear inequalities represent a region on a coordinate plane.

What is a Linear Inequality?

A linear inequality in two variables (typically x and y) is an inequality that can be written in the form:

• $Ax + By < C$
• $Ax + By > C$
• $Ax + By \le C$
• $Ax + By \ge C$

where A, B, and C are constants, and A and B are not both zero. The symbols , ≤, and ≥ indicate the type of inequality.

How to Graph a Linear Inequality

Graphing a linear inequality involves two main steps:

1. Graph the Boundary Line: First, treat the inequality as an equation (e.g., $Ax + By = C$) and graph this line.
   • If the inequality is strict (< or >), the line is dashed, indicating that points on the line are not part of the solution set.
   • If the inequality includes "or equal to" (≤ or ≥), the line is solid, indicating that points on the line are part of the solution set.
   A common way to graph the boundary line is by finding its x- and y-intercepts.
   • To find the x-intercept, set $y=0$ and solve for $x$.
   • To find the y-intercept, set $x=0$ and solve for $y$.
2. Shade the Solution Region: After graphing the boundary line, you need to determine which side of the line represents the solution set. This is done by choosing a test point that is not on the line (the origin (0,0) is often the easiest, unless the line passes through it).
   • Substitute the coordinates of the test point into the original inequality.
   • If the inequality is true, shade the region that contains the test point.
   • If the inequality is false, shade the region that does not contain the test point.
   For inequalities in the form $Ax + By < C$ or $Ax + By \le C$, the shading is typically below the line if $B > 0$, or above the line if $B C$ or $Ax + By \ge C$, it's typically above the line if $B > 0$, or below the line if $B < 0$. For vertical lines ($By = C$), shading is left or right. For horizontal lines ($Ax = C$), shading is above or below.

Calculator Explanation

This calculator helps visualize these steps. You input the coefficients (A, B, C), the variables (x, y), the inequality sign, and the desired shading direction. It then provides a textual description of the boundary line and the region to be shaded, assisting in the graphing process.

Example Usage

Let's graph the inequality $2x + 3y \le 6$.

• Variables: x, y
• Coefficients: A=2, B=3
• Constant: C=6
• Inequality Sign: ≤ (Less Than or Equal To)

Steps:

1. Boundary Line: Graph $2x + 3y = 6$.
   • If $x=0$, then $3y = 6$, so $y = 2$. The y-intercept is (0, 2).
   • If $y=0$, then $2x = 6$, so $x = 3$. The x-intercept is (3, 0).
   Since the inequality is ≤, the line will be solid. Draw a solid line through (0, 2) and (3, 0).
2. Shading: Test the point (0,0). $2(0) + 3(0) \le 6$ $0 \le 6$ This is true. Therefore, shade the region containing (0,0), which is below the line.

The calculator output will guide you through these specific instructions based on your inputs.