Materials Needed

• Notebook or digital tablet for calculations
• Graph paper or a graphing calculator (Desmos is a great digital option)
• Colored pens or highlighters
• "The Side Hustle" scenario worksheet (provided in the activity section)

Learning Objectives

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

• Translate real-world scenarios into 1-variable linear inequalities.
• Solve linear inequalities using algebraic properties.
• Represent solutions on a number line.
• Explain why the inequality sign flips when multiplying or dividing by a negative number.

1. Introduction: The "Not-Exactly" Reality (Hook)

In math class, we spend a lot of time finding x = 5. But in real life, things are rarely that exact. If you have $40 in your pocket for a night out, you don't have to spend exactly $40. You need your total costs to be less than or equal to $40. If you’re driving and the sign says the speed limit is 65, your speed needs to be less than or equal to 65 (unless you want a ticket).

Inequalities are the math of boundaries, budgets, and limits. Today, we’re going to learn how to write these limits as math problems and solve them to find our "zone of success."

2. Content & Practice: The "I Do, We Do, You Do" Model

I Do: The Anatomy of an Inequality

Writing an inequality is just like writing an equation, but we use "boundary symbols" instead of an equals sign:

• < : Less than (fewer than, below)
• > : Greater than (more than, above)
• ≤ : Less than or equal to (at most, maximum)
• ≥ : Greater than or equal to (at least, minimum)

Example Scenario: You want to buy a new pair of headphones that cost $120. You have $30 saved, and you earn $15 an hour washing cars. How many hours (h) do you need to work?

The Setup: 15h + 30 ≥ 120

The Solution: Subtract 30: 15h ≥ 90. Divide by 15: h ≥ 6. Interpretation: You need to work at least 6 hours.

The Golden Rule (The Negative Flip)

There is one major difference between equations and inequalities. If you multiply or divide by a negative number, you must flip the sign. Why? Think about it: 5 > 2. If we multiply both by -1, we get -5 and -2. Is -5 greater than -2? No! So we must flip it: -5 < -2.

We Do: The "Bad Subscription" Logic

Let's look at this together. Imagine you are cancelling a subscription. They owe you a refund, but they charge a "processing fee."

Scenario: -3x + 12 < 3

1. First, what's our first step? (Subtract 12 from both sides).
2. Result: -3x < -9.
3. Now, we divide by -3. What happens to the sign? (It flips!).
4. Solution: x > 3.

You Do: The Side Hustle Challenge

Now it’s your turn to apply this to a real-world project. Choose one of the following scenarios to solve:

• Scenario A (The Sneaker Reseller): You buy sneakers for $100 and want to sell them on an app. The app takes a flat $15 fee. You want to make a profit of more than $50. Write and solve an inequality to find the minimum price (p) you should list.
• Scenario B (The Concert Trip): You and your friends are renting a van for $200. You also need $40 per person for tickets. Your total budget is $500. How many people (p) can go?

3. Hands-On Activity: The Number Line Visualizer

Solving the math is only half the battle; you have to visualize the solution set. Using your solution from the "You Do" activity above:

1. Draw a number line.
2. Place an open circle if the sign is < or > (because the boundary number isn't included).
3. Place a closed circle if the sign is ≤ or ≥ (because the boundary number is included).
4. Shade the line in the direction of the solution.

Self-Check: Pick a number in your shaded area and plug it back into your original inequality. Does the math still work? If yes, you nailed it.

4. Conclusion: Recap & Real-World Mastery

Today we moved beyond the "equals sign" to handle the math of reality—inequalities. Here is the "cheat sheet" of what we covered:

• Identify the Goal: Look for keywords like "at least" or "maximum" to pick your sign.
• Isolate the Variable: Use the same inverse operations you use for regular equations.
• The Flip: Never forget to flip that sign if you multiply or divide by a negative!
• The Shading: A solution isn't just one number; it's a whole range of possibilities.

5. Assessment: Show What You Know

Formative Assessment (Quick Check)

Explain in your own words why the inequality x > 5 has infinitely many solutions, while x = 5 only has one.

Summative Assessment (The Final Task)

Create your own inequality word problem based on something you want to buy or a goal you want to achieve this month.

1. Write the inequality.
2. Show the step-by-step solution.
3. Graph the solution on a number line.
4. Write one sentence explaining what the answer means for your goal.

6. Differentiation & Extensions

• Scaffolding (For a smoother start): Use a digital "Equation Balancer" or virtual tiles to visualize moving numbers across the inequality sign.
• Extension (For a challenge): Try a compound inequality. Example: You want to spend more than $20 but no more than $50 on a gift. How does that look on a graph? (Hint: It’s a "sandwich" graph!)

Success Criteria

• [ ] I can identify the correct inequality symbol from a word problem.
• [ ] I can solve for x by performing operations on both sides.
• [ ] I remember to flip the sign when dividing/multiplying by negatives.
• [ ] I can graph the solution with the correct circle type (open/closed) and shading.