Mastering the Hustle: Writing and Graphing Two-Variable Linear Inequalities
Lesson Overview
This lesson transforms abstract algebra into a practical tool for decision-making. Students will learn to translate real-world constraints—like budgets and time limits—into linear inequalities, graph them to visualize "possibility zones," and identify winning strategies.
Materials Needed
- Graph paper (or a digital graphing tool like Desmos)
- Ruler or straightedge
- Two different colored highlighters or colored pencils
- Calculator
- "The Side Hustle Challenge" worksheet (included in activities)
Learning Objectives
By the end of this lesson, you will be able to:
- Identify key phrases that translate to inequality symbols (<, >, ≤, ≥).
- Write a two-variable linear inequality from a real-world scenario.
- Graph the boundary line of an inequality (distinguishing between solid and dashed lines).
- Shade the correct "half-plane" to represent all possible solutions.
1. The Hook: The Budget Constraint (5-10 mins)
Scenario: You are planning a weekend road trip. You have exactly $200 to spend on gas (x) and food (y). You know you'll spend more on food than gas because you’re a foodie, but you can’t spend more than the $200 total.
Think-Pair-Share (or Journal Prompt): If you spend $150 on gas, how much can you spend on food? If you spend $50 on gas, what is the maximum you can spend on food? Is there only one "right" answer, or a whole range of possibilities?
Key takeaway: In the real world, we rarely have one "equal" answer. We usually have a range of options limited by a "boundary." That range is an inequality.
2. Instruction: The "I Do" Model (15 mins)
Step 1: The Anatomy of the Inequality
To write an inequality, we look for "Constraint Language":
| Phrase | Symbol | Graph Line Type |
|---|---|---|
| "Less than," "Below" | < | Dashed (Boundary not included) |
| "More than," "Exceeds" | > | Dashed (Boundary not included) |
| "At most," "Maximum of," "No more than" | ≤ | Solid (Boundary is included) |
| "At least," "Minimum of," "No less than" | ≥ | Solid (Boundary is included) |
Step 2: From Words to Math
Example: "You are selling custom t-shirts (x) for $20 and hoodies (y) for $40. You need to make at least $800 to cover your costs."
Mathematical Translation: 20x + 40y ≥ 800
Step 3: The Visualization (Graphing)
- Find the intercepts: Set x=0 to find the y-intercept (0, 20). Set y=0 to find the x-intercept (40, 0).
- Draw the line: Since it is ≥, use a solid line.
- The Test Point: Pick (0,0). Is 20(0) + 40(0) ≥ 800? No (0 is not ≥ 800). Since (0,0) failed, we shade the side of the line away from the origin.
3. Guided Practice: The "We Do" Model (15 mins)
Let's solve this together. Imagine you are a social media manager. You charge $50/hour for content creation (x) and $75/hour for ad management (y). You want to work no more than 30 hours a week, but you need to earn more than $1,500.
Task: Let's focus on the earnings inequality.
- Identify the variables: x = content hours, y = ad hours.
- Write the inequality: 50x + 75y > 1500.
- The Boundary: Is it solid or dashed? (Student answer: Dashed, because it's "greater than," not "equal to.")
- Plotting: If x is 30, what is y? (50*30 + 75y = 1500 -> 1500 + 75y = 1500 -> y = 0). Point: (30,0). If y is 20, what is x? (50x + 75*20 = 1500 -> 50x + 1500 = 1500 -> x = 0). Point: (0,20).
- Shading: If we work 40 hours of content (40, 10), do we make more than $1500? (50*40 + 75*10 = 2000 + 750 = 2750). Yes! 2750 > 1500. Shade the region containing (40, 10).
4. Independent Practice: The "You Do" Challenge (20 mins)
The Scenario: The Concert Promoter
You are promoting a local show. General admission tickets (x) are $15 and VIP tickets (y) are $35. The venue has a maximum capacity of 500 people. You must make at least $7,000 to pay the band and the venue.
Your Task:
- Write an inequality representing the revenue (money made).
- Write an inequality representing the capacity (number of people).
- Graph both on the same coordinate plane.
- Pick three "coordinate points" (x,y) in your shaded region and explain what they represent in real life (e.g., "Selling 200 GA tickets and 200 VIP tickets").
- Identify one point that is not a solution and explain why (e.g., "Selling 600 GA tickets—it makes enough money, but the building isn't big enough!").
5. Closure and Recap (5 mins)
- Recap: We learned that inequalities represent ranges of possibilities. "At least" means ≥ (solid line), and "Less than" means < (dashed line).
- The "Aha!" Moment: Ask the student: "Why is the shading usually more important than the line itself in business?" (Answer: Because the shading represents all the different ways you can succeed/be profitable).
- Success Criteria Check: Can you look at a graph and tell if the boundary is included? Can you explain what a point in the shaded region means for a business owner?
Assessment Methods
Formative (During lesson): Thumbs up/down on whether a phrase like "no more than" results in a solid or dashed line. Correctly identifying the test point (0,0) results.
Summative (End of lesson): The "Concert Promoter" graph.
- Level 1 (Developing): Can write the inequality but struggles with graphing.
- Level 2 (Proficient): Can write the inequality and graph with correct line types and shading.
- Level 3 (Mastery): Can interpret points in the shaded region and explain why certain points (like negatives) don't make sense in a real-world context.
Differentiation & Adaptability
- For the Tech-Savvy: Use Desmos or GeoGebra to graph the inequalities. Use the "sliders" feature to see how changing the price of a ticket shifts the "possibility zone."
- For the Kinesthetic Learner: Use a large coordinate grid on the floor with masking tape. Have the student stand in the "Solution Zone" vs. the "Failure Zone."
- Extension (Advanced): Introduce a third constraint (e.g., "You must sell at least 50 VIP tickets for the sponsors to be happy") and find the overlapping region (System of Inequalities).