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Graphing Systems of Linear Inequalities

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Graphing Systems of Linear Inequalities

Study Guide

Key Definition

The solution of a system of linear inequalities is the intersection of the regions defined by each inequality.

Important Notes

Graph the line of the equation by replacing the inequality with an equality.
Use a solid line for $ \leq $ or $ \geq $ and a dashed line for $ < $ or $ > $.
Shade the half-plane that satisfies the inequality.
The intersection of shaded regions is the solution.
Use a test point like $ (0, 0) $ to determine shading.

Mathematical Notation

$\leq$ means less than or equal to

$\geq$ means greater than or equal to

$+$ represents addition

Remember to use proper notation when solving problems

Why It Works

By graphing each inequality, we visually find the common region that satisfies all conditions in a system.

Remember

Always check your shaded region with a test point to ensure you shaded the correct half-plane.

Quick Reference

Solid Line:$\leq \text{ or } \geq$

Dashed Line:$< \text{ or } >$

Understanding Graphing Systems of Linear Inequalities

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

To graph a system of inequalities, first graph each boundary line by rewriting the inequality as an equation. Use a solid line for ≤ or ≥ and a dashed line for < or >. Then choose a test point not on the line (for example (0,0)) to determine which side satisfies each inequality and shade that half-plane.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following inequalities is graphed with a solid boundary line?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have a budget of $50 to buy x items costing $5 each and y items costing $3 each: $5x + 3y \leq 50$.

Thinking Challenge

Thinking Exercise