Real World Examples of Systems of Linear Inequalities Explained

📚 Quick Study Guide

• 🍎 Linear Inequality Basics: A linear inequality compares two expressions using inequality symbols like < (less than), > (greater than), $\leq$ (less than or equal to), or $\geq$ (greater than or equal to).
• 📈 Graphing Linear Inequalities: Graph the related linear equation. Use a solid line for $\leq$ or $\geq$ and a dashed line for < or >. Shade the region that satisfies the inequality.
• 🍕 Real-World Applications: Linear inequalities can model constraints in resource allocation, budgeting, and optimization problems.
• 💡 Solving Word Problems: Identify the variables, translate the problem into mathematical inequalities, and solve for the unknowns.

Practice Quiz

1. What real-world scenario can be modeled using a system of linear inequalities?
   1. A. Calculating the area of a circle.
   2. B. Determining the trajectory of a projectile.
   3. C. Optimizing a diet plan with minimum nutrient requirements and maximum calorie intake.
   4. D. Finding the roots of a quadratic equation.
2. A student is planning to buy notebooks and pens. Notebooks cost $2 each, and pens cost $1 each. The student has a maximum of $10 to spend. Which inequality represents this situation, where $x$ is the number of notebooks and $y$ is the number of pens?
   1. A. $x + y \leq 10$
   2. B. $2x + y \leq 10$
   3. C. $x + 2y \leq 10$
   4. D. $2x + 2y \leq 10$
3. A bakery makes cakes and cookies. A cake requires 2 cups of flour and a batch of cookies requires 1 cup of flour. The bakery has 12 cups of flour available. If $x$ represents the number of cakes and $y$ represents the number of batches of cookies, which inequality represents the flour constraint?
   1. A. $x + y \leq 12$
   2. B. $2x + y \leq 12$
   3. C. $x + 2y \leq 12$
   4. D. $2x + 2y \leq 12$
4. A farmer wants to plant at least 50 acres of corn and soybeans. Let $x$ be the number of acres of corn and $y$ be the number of acres of soybeans. Which inequality represents this constraint?
   1. A. $x + y \leq 50$
   2. B. $x + y \geq 50$
   3. C. $x - y \geq 50$
   4. D. $x - y \leq 50$
5. A company produces two types of products, A and B. Product A requires 2 hours of labor, and product B requires 3 hours of labor. There are at most 24 hours of labor available. If $x$ represents the number of units of product A and $y$ represents the number of units of product B, which inequality represents the labor constraint?
   1. A. $2x + 3y \geq 24$
   2. B. $3x + 2y \leq 24$
   3. C. $2x + 3y \leq 24$
   4. D. $3x + 2y \geq 24$
6. A health-conscious individual wants to consume at least 60 grams of protein and at least 40 grams of carbohydrates daily. Let $x$ be the grams of protein and $y$ be the grams of carbohydrates. Which of the following represents the system of inequalities?
   1. A. $x \geq 60, y \geq 40$
   2. B. $x \leq 60, y \leq 40$
   3. C. $x + y \geq 100$
   4. D. $x \geq 40, y \geq 60$
7. A clothing store sells shirts for $10 and pants for $20. They need to make at least $500 in sales each day. If $x$ is the number of shirts and $y$ is the number of pants sold, which inequality represents the sales target?
   1. A. $10x + 20y \leq 500$
   2. B. $20x + 10y \geq 500$
   3. C. $10x + 20y \geq 500$
   4. D. $20x + 10y \leq 500$