#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!

#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)

#Solving Linear Inequality Word Problems

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