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Graphing Linear Inequalities in Two Variables

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Graphing Linear Inequalities in Two Variables

Study Guide

Key Definition

To graph a linear inequality in two variables, express it as $y < mx + b$, $y &leq mx + b$, $y > mx + b$, or $y &geq mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Important Notes

A dashed line is used for $<$ or $>$.
A solid line is used for $\leq$ or $\geq$.
Test a point to determine which half-plane to shade.
The line itself is not included in the solution for $<$ or $>$.
The line is included in the solution for $\leq$ or $\geq$.

Mathematical Notation

$<$ means less than

$>$ means greater than

$\leq$ represents less than or equal to

$\geq$ represents greater than or equal to

$y = mx + b$ is the slope-intercept form

Remember to use proper notation when solving problems

Why It Works

Graphing an inequality shows all possible solutions. Draw the boundary line (solid for ≤/≥, dashed for < />) and shade the region where the inequality holds: above the line for $y > mx + b$ or $y &geq mx + b$, and below for $y < mx + b$ or $y &leq mx + b$.

Remember

Always choose a test point not on the line, such as $(0,0)$, to determine the correct half-plane to shade.

Quick Reference

Slope-Intercept Form:$y = mx + b$

Understanding Graphing Linear Inequalities in Two Variables

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A linear inequality in two variables describes a half-plane. Start by graphing the line y = mx + b (use a solid line for ≤/≥, dashed for < />), then test a point to decide which side to shade.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What type of line is used for $y > 2x + 3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Alex wants to graph the inequality $y \geq -\frac{1}{2}x + 5$ to determine the area of values that meet his budget for a new skateboard.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine which region to shade for $3x - 2y < 6$ using a test point.

Challenge Quiz

Single Choice Quiz

Advanced

Select the correct graph representation for $y &leq \frac{3}{4}x - 2$

Recap

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Review key concepts and takeaways