#Linear Inequalities: Your Guide to Crushing Word Problems 💪

Hey there, future AP superstar! Let's break down linear inequalities in word problems. Think of them as puzzles where you're setting boundaries, not just finding one right answer. Ready to unlock the secrets? Let's dive in!

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Understanding the Basics

#Identifying and Interpreting Inequalities

Linear inequality word problems are all about real-world constraints. They use math to set limits. Here's how to spot them:

• Variables: First, identify what you're trying to find (like the number of items or miles). Assign it a variable (usually x or y).
• Key Phrases: These phrases are your best friends. They tell you which inequality symbol to use:
  • "At least" or "no less than" ➡️ ≥ (greater than or equal to)
  • "At most" or "no more than" ➡️ ≤ (less than or equal to)
  • "More than" ➡️ > (greater than)
  • "Less than" ➡️ < (less than)
• Constraints: Look for any extra limits, like ratios or specific ranges. These also form inequalities.
• Translation: Turn the words into a math expression. Use the variable, numbers, and the correct inequality symbol.

#Examples of Linear Inequalities

Let's make this real with some examples:

• Example 1: A company must produce at least 500 units to break even.
  • Inequality: $x ≥ 500$ (x is the number of units)
• Example 2: The maximum weight limit for luggage is 50 pounds.
  • Inequality: $w ≤ 50$ (w is the weight of luggage)
• Example 3: A car's fuel efficiency should be greater than 30 miles per gallon.
  • Inequality: $e > 30$ (e is the fuel efficiency)
• Example 4: The temperature must remain below 32°F for snow to form.
  • Inequality: $t < 32$ (t is the temperature)

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Solving Linear Inequality Word Problems

#Algebraic Methods

Time to solve! Here's how:

1. Isolate the Variable: Use algebra to get the variable by itself on one side of the inequality.
2. Same Operation: Whatever you do to one side, do to the other (add, subtract, multiply, or divide).
3. Key Concept

Flip the Sign: If you multiply or divide by a negative number, you must reverse the inequality sign.

💡 4. Solution Interval: Your answer will be a range of values that make the inequality true. 5. Real-World Check: Interpret your solution in the context of the problem. Round to the nearest whole number if needed.

#Graphing Techniques

Visualizing inequalities can be super helpful:

• Number Line: Draw a number line.
• Open/Closed Circle:
  • Use an open circle (o) for < or > (the endpoint is not included).
  • Use a closed circle (●) for ≤ or ≥ (the endpoint is included).
• Shading: Shade the part of the number line that includes the solutions. Shade to the right for greater than, and to the left for less than.
• Multiple Inequalities: If you have more than one, find the overlapping shaded area.

Caption: Example of x > 3 on number line, with an open circle at 3 and shading to the right.

#Problem-Solving Examples

Let's tackle a couple of real-world problems:

• Example 1: Profit Calculation
  • Problem: A factory produces widgets at a cost of $5 per unit with a fixed overhead of$1000. The selling price is $8 per unit. How many units must be sold to make a profit?$
  • Solution:
    • Let x be the number of units sold.
    • Revenue:8x$* Cost:$5x + 1000$* Profit occurs when revenue > cost:$8x > 5x + 1000$* Solve:$3x > 1000$,$x > 333.33$* [object Object] The factory must sell at least 334 units to make a profit.$
• Example 2: Car Rental Budget
  • Problem: A car rental company charges40 per day plus $0.25 per mile driven. If a customer's budget is$200, how many miles can they drive in 3 days?
  • Solution:
    • Let m be the number of miles driven.
    • Total cost: $40(3) + 0.25m ≤ 200$
    • Solve: $120 + 0.25m ≤ 200$, $0.25m ≤ 80$, $m ≤ 320$
    • Interpretation: The customer can drive up to 320 miles within their budget.

#Final Exam Focus

Alright, let's get down to the nitty-gritty. Here's what you should really focus on:

• High-Priority Topics:
  • Translating word problems into inequalities.
  • Solving inequalities using algebraic methods.
  • Graphing inequalities on a number line.
  • Interpreting solutions in the context of the problem.
• Common Question Types:
  • Profit/loss scenarios.
  • Budgeting problems.
  • Distance/rate/time problems with constraints.
  • Mixture problems with quantity limits.
• Last-Minute Tips:
  • Read Carefully: Understand the context before jumping into the math.
  • Show Your Work: Even if you make a small mistake, you can get partial credit.
  • Check Your Answer: Does your solution make sense in the real world?
  • Time Management: Don't spend too long on one problem. If you're stuck, move on and come back to it later.
  • Don't Panic: You've got this! Take a deep breath and trust your preparation.

#Practice Questions

Okay, let's put your knowledge to the test with some practice questions. Remember, practice makes perfect!

Practice Question

Multiple Choice Questions

1. A student needs to earn at least 80 points on a test to get a B. If each question is worth 4 points, what is the minimum number of questions the student needs to answer correctly? (A) 18 (B) 19 (C) 20 (D) 21

2. A bakery makes cookies. The cost to make each cookie is $0.50, and the bakery has a fixed daily cost of$50. If the cookies are sold for $1.50 each, what is the minimum number of cookies that must be sold each day to make a profit? (A) 49 (B) 50 (C) 51 (D) 52$

Free Response Question

A company makes two types of products: widgets and gadgets. The company can make at most 1000 products in a day. Each widget costs5 to make and sells for $12. Each gadget costs$8 to make and sells for $20. The company wants to make at least$10,000 in revenue per day. Let x be the number of widgets and y be the number of gadgets.

(a) Write an inequality that represents the constraint on the total number of products. (b) Write an inequality that represents the constraint on the daily revenue. (c) If the company decides to make 400 widgets, what is the minimum number of gadgets they need to make to meet their revenue goal?

Scoring Breakdown

(a) 1 point: $x + y ≤ 1000$ (b) 2 points: $12x + 20y ≥ 10000$ (c) 3 points: * Substitute $x = 400$ into the revenue inequality: $12(400) + 20y ≥ 10000$ * Simplify: $4800 + 20y ≥ 10000$ * Solve for y: $20y ≥ 5200$, $y ≥ 260$. Minimum number of gadgets is 260.

That's it! You're now equipped to tackle any linear inequality word problem that comes your way. Keep practicing, stay confident, and you'll ace that AP exam. You've got this! 🎉