Lesson 86: Solving Inequalities

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

This lesson introduces solving linear inequalities. Students will learn how to manipulate inequalities similarly to equations, with attention to rules for reversing the inequality sign when multiplying or dividing by negative numbers. Solutions will be expressed using inequality notation and number lines.

Learning Objectives

• Solve linear inequalities using addition, subtraction, multiplication, and division.
• Understand when to reverse the inequality sign when multiplying or dividing by a negative number.
• Represent solution sets using inequality notation and number lines.

Topics Covered

• One-step and multi-step inequalities
• Multiplying/dividing by negative numbers
• Solution sets and number line representation
• Compound inequalities

Key Terms

• Inequality: A mathematical statement that compares two expressions using symbols such as <, >, ≤, ≥.
• Solution Set: The set of all values that satisfy an inequality.
• Compound Inequality: Two inequalities joined together, often using "and" or "or".

Lesson Examples

Example 1: Solving a One-Step Inequality

1. Solve: x + 5 > 12
2. Subtract 5 from both sides: x > 7
3. Solution: All numbers greater than 7. Represent on number line.

Example 2: Multiplying by a Negative Number

1. Solve: -3x < 9
2. Divide both sides by -3 (reverse inequality sign): x > -3
3. Solution: All numbers greater than -3.

Practice Problems

1. 2x - 4 < 6
2. -5x + 3 > 8
3. 3(x - 2) ≤ 9
4. -2(x + 1) ≥ 6

Tips & Common Mistakes

• Always reverse the inequality sign when multiplying or dividing by a negative number.
• Check solutions by substituting a value from the solution set into the original inequality.
• Use number lines to visualize solution sets and avoid mistakes with compound inequalities.

Summary

Solving inequalities uses similar steps as equations, with the key difference being the reversal of the inequality sign when multiplying or dividing by a negative. Representing solutions clearly using inequality notation and number lines ensures accurate understanding and communication of results.

Challenge Problems

1. 4x - 7 < 9
2. -3(x + 2) > 6
3. 5x + 1 ≤ 2x + 10