Lesson 6: Linear Inequalities and Graphical Solutions

👨🏽‍🏫 Introduction

Suppose you want to find out how many items you can buy without spending more than ₦1,000, or how many hours you need to study to get at least 60% on a test. ⏱️💰 These are situations where inequalities — not equations — help us find answers.

In this lesson, you’ll learn how to solve and represent linear inequalities both algebraically and graphically.

By the end of this lesson, you’ll be able to:

• Solve linear inequalities in one or two variables
• Represent solutions on number lines and graphs
• Shade and interpret solution regions on graphs

🧠 Key Concepts and Explanations

🔢 What is a Linear Inequality?

A linear inequality looks like a linear equation, but it uses inequality signs such as:

• < (less than)
• ≤ (less than or equal to)
• > (greater than)
• ≥ (greater than or equal to)

Examples:

📏 Solving Linear Inequalities in One Variable

1. Solve the inequality like an equation.
2. If you multiply or divide both sides by a negative number, reverse the inequality sign.

Example:

📉 Representing Solutions on a Number Line

• Use a hollow circle (◯) for < or >
• Use a filled circle (●) for ≤ or ≥
• Draw an arrow in the direction of valid solutions

🧭 Linear Inequalities in Two Variables

These involve both x and y. Examples:

To graph an inequality in two variables:

1. Convert the inequality to an equation (e.g., y = 2x + 3).
2. Draw a solid line for ≤ or ≥, and a dashed line for < or >.
3. Pick a test point (usually (0,0)) and check if it satisfies the inequality.
4. Shade the correct region based on the result.

🧮 Sample Problem Walkthroughs

1. Solve and represent: 2x – 5 < 7
   

   

   

   Represent this on a number line with a hollow circle at 6 and an arrow pointing left.

2. Graph the inequality: y ≥ x – 2
   • Plot the line .
   • Use a solid line because the inequality includes “equal to”.
   • Pick (0, 0) as a test point. Since 0 ≥ -2 is true, shade above the line.
3. Graph the inequality: y < -x + 3
   • Plot the line .
   • Use a dashed line because the inequality is strict (<).
   • Test point (0, 0): Since 0 < 3 is true, shade below the line.

✍🏽 Practice Exercises

🅰️ A. One Variable

1. Solve and graph:
2. Solve and graph:

🅱️ B. Two Variables

1. Graph the inequality:
2. Graph the inequality:

🆎 C. Combined Regions

1. Sketch the region that satisfies both and

✅ Answers and Explanations

🅰️ A. One Variable

• 1. → → Filled circle at 2, arrow right.
• 2. → → Filled circle at -3, arrow right.

🅱️ B. Two Variables

• 1. Line is . Use a solid line and shade below.
• 2. Line is . Use a dashed line and shade above.

🆎 C. Combined Regions

• The solution region is between and . Shade the overlapping area.

📌 Recap

• ✅ Linear inequalities describe a range of values instead of a fixed number.
• ✅ You can solve and graph inequalities in one or two variables.
• ✅ Use solid or dashed lines and shading to show valid regions on a graph.

💭 Reflection Prompt

Can you think of a real-world situation where two or more conditions (inequalities) must be satisfied at the same time? How would you represent that on a graph?