Master

Systems of Linear Inequalities

Master systems of linear inequalities with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Systems of Linear Inequalities

Study Guide

Key Definition

A system of linear inequalities consists of two or more linear inequalities in the same variables. The solution is the region where the graphs of all inequalities overlap.

Important Notes

Use a solid line for $\leq$ or $\geq$
Use a dashed line for $<$ or $>$
Shade above the line for $\geq$ or $>$
Shade below the line for $\leq$ or $<$
Check by testing a point not on the boundary

Mathematical Notation

$\leq$ means less than or equal to

$\geq$ means greater than or equal to

$<$ represents less than

$>$ represents greater than

Remember to use proper notation when solving problems

Why It Works

When you graph each linear inequality, you get a half-plane. The solution to a system is the set of points common to all half-planes, called the feasible region. This region satisfies every inequality simultaneously.

Remember

Always test a point to determine the correct region to shade for inequalities.

Quick Reference

Solid Line:Use for $\leq$ and $\geq$

Dashed Line:Use for $<$ and $>$

Understanding Systems of Linear Inequalities

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Graph each inequality by drawing its boundary line (solid for $\leq$ or $\geq$, dashed for $<$ or $>$), then shade the half-plane that satisfies the inequality. The solution to the system is the overlapping shaded region.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following lines should be dashed for $y < 2x + 1$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A teenager wants to save money and decides he can spend less than $10 per day. How can we express this as an inequality?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

On a coordinate plane, graph the line $y = -3x - 1$ as a solid boundary. Which region should be shaded to represent the solution of $y \geq -3x - 1$?

Challenge Quiz

Single Choice Quiz

Advanced

Which system of inequalities represents the region shaded below the line $y = \frac{1}{3}x + 4$ (dashed boundary) and above the line $y = 5x + 1$ (solid boundary)?

Recap

Watch & Learn

Review key concepts and takeaways