Linear Inequalities

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Answer Key & Explanations

Linear Inequalities · Grade 8 · Worksheet 1

1. Liam is organizing a school fundraiser and needs to buy supplies. He has a budget of $120 to spend on two types of items: pens that cost $2 each and notebooks that cost $5 each. He needs at least 15 pens and at least 10 notebooks. Write an inequality that represents the situation where Liam does not exceed his budget, using x for pens and y for notebooks. Answer: 2x + 5y ≤ 120 Solution: - \( x \) = number of pens - \( y \) = number of notebooks - Each pen costs $2, so total cost for pens = \( 2 \times x \) = \( 2x \) - Each notebook costs $5, so total cost for notebooks = \( 5 \times y \) = \( 5y \) 2x + 5y Liam has a budget of $120.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Identify the variables** We are told: - \( x \) = number of pens - \( y \) = number of notebooks --- **Step 2: Identify the costs per item** - Each pen costs $2, so total cost for pens = \( 2 \times x \) = \( 2x \) - Each notebook costs $5, so total cost for notebooks = \( 5 \times y \) = \( 5y \) --- **Step 3: Total cost expression** The total cost is: \[ 2x + 5y \] --- **Step 4: Budget constraint** Liam has a budget of $120. "Does not exceed his budget" means the total cost must be less than or equal to 120. So: \[ 2x + 5y \leq 120 \] --- **Step 5: Check other conditions** The problem says he needs at least 15 pens and at least 10 notebooks. That means \( x \geq 15 \) and \( y \geq 10 \). But the question only asks for the inequality that represents not exceeding the budget, so we don't include \( x \geq 15 \) and \( y \geq 10 \) in this particular inequality. --- **Step 6: Final answer** The inequality for the budget is: \[ 2x + 5y \leq 120 \]

2. Is (9, 20) a solution to y > 3x - 7? Answer: Yes Solution: Substitute x = 9 and y = 20 into the inequality y > 3x - 7. Left side: y = 20. Right side: 3(9) - 7 = 27 - 7 = 20.
   Full step-by-step solution

   Step 1: Substitute x = 9 and y = 20 into the inequality y > 3x - 7. Step 2: Left side: y = 20. Step 3: Right side: 3(9) - 7 = 27 - 7 = 20. Step 4: Check the inequality: 20 > 20 is false because 20 is not greater than 20. Step 5: Since the inequality is false, the point (9, 20) is not a solution. The answer is No.

3. Emma is planning a community garden and needs to buy soil and fertilizer. She has a budget of $200. Soil bags cost $8 each and fertilizer bags cost $12 each. She needs at least 15 bags of soil for the garden beds and wants to buy at least 5 bags of fertilizer. Write a system of inequalities that represents Emma's situation, where s is the number of soil bags and f is the number of fertilizer bags. Answer: 8s + 12f ≤ 200, s ≥ 15, f ≥ 5 Solution: Identify the cost constraint. Each soil bag costs $8 and each fertilizer bag costs $12, so the total cost is 8s + 12f. Since Emma's budget is $200, we write: 8s + 12f ≤ 200 Identify the minimum soil requirement.
   Full step-by-step solution

   Step 1: Identify the cost constraint. Each soil bag costs $8 and each fertilizer bag costs $12, so the total cost is 8s + 12f. Since Emma's budget is $200, we write: 8s + 12f ≤ 200 Step 2: Identify the minimum soil requirement. Emma needs at least 15 bags of soil, so: s ≥ 15 Step 3: Identify the minimum fertilizer requirement. Emma wants at least 5 bags of fertilizer, so: f ≥ 5 Step 4: Combine all constraints into a system of inequalities: 8s + 12f ≤ 200 s ≥ 15 f ≥ 5 The complete system is: 8s + 12f ≤ 200, s ≥ 15, f ≥ 5

4. Hana is selling two types of handmade jewelry at a school fair. She makes beaded necklaces that sell for $15 each and wire-wrapped earrings that sell for $12 per pair. She wants to earn at least $240 in total sales. However, she can make at most 25 items total because of the time she has available. Write a system of inequalities to represent this situation, using x for the number of necklaces and y for the number of pairs of earrings. Then, determine if she can sell 12 necklaces and 10 pairs of earrings while meeting both of her goals. Answer: No Solution: Write the inequality for the total sales goal. Each necklace sells for $15, so x necklaces bring in 15x dollars. Each pair of earrings sells for $12, so y pairs bring in 12y dollars.
   Full step-by-step solution

   Step 1: Write the inequality for the total sales goal. Each necklace sells for $15, so x necklaces bring in 15x dollars. Each pair of earrings sells for $12, so y pairs bring in 12y dollars. She wants at least $240, so: 15x + 12y >= 240. Step 2: Write the inequality for the total number of items. She can make at most 25 items total, so: x + y <= 25. Step 3: Test the point (12, 10) where x = 12 necklaces and y = 10 pairs of earrings. Check the sales inequality: 15(12) + 12(10) = 180 + 120 = 300. Is 300 >= 240? Yes, this condition is met. Check the item limit inequality: 12 + 10 = 22. Is 22 <= 25? Yes, this condition is also met. Since both inequalities are satisfied, Hana can sell 12 necklaces and 10 pairs of earrings while meeting both of her goals. The answer is Yes.

5. 2x + 3y ≤ 12; x ≥ 0; y ≥ 0; Find the maximum value of P = 3x + 2y Answer: 18 Solution: 2x + 3y ≤ 12 x ≥ 0 y ≥ 0 Objective function: P = 3x + 2y Find the corner points of the feasible region.
   Full step-by-step solution

   We are given the constraints: 2x + 3y ≤ 12 x ≥ 0 y ≥ 0 Objective function: P = 3x + 2y --- **Step 1: Find the corner points of the feasible region.** The line 2x + 3y = 12 intersects the axes at: - When y = 0: 2x = 12 → x = 6 → point (6, 0) - When x = 0: 3y = 12 → y = 4 → point (0, 4) Also, the constraints x ≥ 0, y ≥ 0 mean we are in the first quadrant. So the feasible region is a triangle with vertices: (0, 0), (6, 0), (0, 4). --- **Step 2: Evaluate P = 3x + 2y at each vertex.** At (0, 0): P = 3(0) + 2(0) = 0 At (6, 0): P = 3(6) + 2(0) = 18 At (0, 4): P = 3(0) + 2(4) = 8 --- **Step 3: Check if maximum occurs at a vertex (by linear programming theory).** Since the objective function is linear and the feasible region is a convex polygon, the maximum occurs at a vertex. The largest value among 0, 18, 8 is 18. --- **Step 4: Conclusion** Maximum value of P is 18 at (6, 0). --- **Final answer:** 18

6. Noah is planning a community garden and needs to buy soil and fertilizer. He has a budget of $200. Soil costs $4 per bag and fertilizer costs $8 per container. Noah needs at least 25 bags of soil for the garden beds. Write an inequality that represents all possible combinations of soil bags (s) and fertilizer containers (f) that Noah can purchase while staying within his budget and meeting the minimum soil requirement. Answer: 4s + 8f ≤ 200, s ≥ 25 Solution: Identify the cost per item: soil costs $4 per bag, fertilizer costs $8 per container Write the cost inequality: 4s + 8f ≤ 200 Write the minimum requirement: s ≥ 25 Combine both conditions: 4s + 8f ≤ 200 and s ≥ 25 The system of inequalities is: 4s + 8f ≤ 200, s ≥ 25
   Full step-by-step solution

   Step 1: Identify the cost per item: soil costs $4 per bag, fertilizer costs $8 per container Step 2: Write the cost inequality: 4s + 8f ≤ 200 Step 3: Write the minimum requirement: s ≥ 25 Step 4: Combine both conditions: 4s + 8f ≤ 200 and s ≥ 25 The system of inequalities is: 4s + 8f ≤ 200, s ≥ 25

7. 2x + 3y ≤ 12 where x = 2 and y = 3 Answer: A. true Solution: Write down the inequality and the given values. Inequality: 2x + 3y ≤ 12 Given: x = 2, y = 3 Substitute the values of x and y into the expression on the left side.
   Full step-by-step solution

   Let's check if the inequality is true when x = 2 and y = 3. Step 1: Write down the inequality and the given values. Inequality: 2x + 3y ≤ 12 Given: x = 2, y = 3 Step 2: Substitute the values of x and y into the expression on the left side. 2x + 3y becomes 2*(2) + 3*(3) Step 3: Perform the multiplication first (following order of operations). 2*(2) = 4 3*(3) = 9 Step 4: Add the results together. 4 + 9 = 13 Step 5: Compare the result (13) to the number on the right side of the inequality (12). We check: Is 13 ≤ 12? Step 6: Analyze the comparison. 13 is not less than or equal to 12. 13 is greater than 12. Therefore, the statement 13 ≤ 12 is false. Step 7: Final conclusion. Since substituting the values makes the inequality false, the correct answer is false. ANSWER: false