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Solving Linear Inequalities

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Solving Linear Inequalities

Study Guide

Key Definition

Inequalities are mathematical sentences comparing two quantities that are not necessarily equal. For example, $x \neq 3$ means $x$ is not equal to 3.

Important Notes

When multiplying or dividing both sides by a negative number, reverse the inequality sign.
Graphing solutions on a number line can help visualize inequalities.
Inequalities like $x \leq 3$ include the number itself.
Use properties of equality to solve inequalities similarly to equations.
Check solutions by substituting back into the original inequality.

Mathematical Notation

$\neq$ means not equal to

$<$ represents less than

$>$ represents greater than

$\leq$ represents less than or equal to

$\geq$ represents greater than or equal to

Remember to use proper notation when solving problems

Why It Works

Using properties of equality such as addition, subtraction, multiplication, and division helps isolate the variable, just as with equations, but remember to reverse the inequality when multiplying or dividing by a negative.

Remember

Always check your solution by substituting the variable back into the original inequality to ensure it holds true.

Quick Reference

Reversing Inequality:When multiplying or dividing by a negative number, reverse the inequality sign.

Understanding Solving Linear Inequalities

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An inequality is like an equation but uses <, >, ≤, or ≥ instead of =. To solve a simple inequality with the variable already isolated, read the symbol and identify all numbers that satisfy it. For example, x ≤ 3 means all x values up to and including 3: {…,-2, -1, 0, 1, 2, 3}. Similarly, x ≥ -4 means all x values from -4 upward. No further steps are required when the coefficient of x is 1.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Solve $2x - 5 < 7$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If you earn $10$ dollars an hour, how many hours must you work to earn at least $100$ dollars?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens if you multiply both sides of $-2x < 4$ by $-1$?

Challenge Quiz

Single Choice Quiz

Advanced

Solve $-3x + 2 \leq 14$

Recap

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