Linear Inequalities

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Answer Key & Explanations

Linear Inequalities · Grade 8 · Worksheet 2

1. Maya is planning a community garden and needs to buy fencing for two rectangular plots. The first plot requires fencing that costs $4 per foot, and the second plot requires fencing that costs $6 per foot. She has a budget of $300 for fencing and needs at least 20 feet of the $6-per-foot fencing for the vegetable plot. Write an inequality that represents all possible combinations of fencing lengths (x for the $4-per-foot fencing, y for the $6-per-foot fencing) that Maya can purchase while staying within her budget and meeting the vegetable plot requirement. Answer: 4x + 6y ≤ 300, y ≥ 20 Solution: Identify the cost components: x feet at $4 per foot costs 4x dollars, y feet at $6 per foot costs 6y dollars. Write the budget constraint: The total cost 4x + 6y must be less than or equal to $300, so 4x + 6y ≤ 300.
   Full step-by-step solution

   Step 1: Identify the cost components: x feet at $4 per foot costs 4x dollars, y feet at $6 per foot costs 6y dollars. Step 2: Write the budget constraint: The total cost 4x + 6y must be less than or equal to $300, so 4x + 6y ≤ 300. Step 3: Write the minimum requirement: Maya needs at least 20 feet of the $6-per-foot fencing, so y ≥ 20. Step 4: Combine both inequalities to form the system: 4x + 6y ≤ 300 and y ≥ 20. The complete system is: 4x + 6y ≤ 300, y ≥ 20.

2. Maya is planning a community garden and needs to buy soil and fertilizer. She has a budget of $200. Soil costs $4 per bag and fertilizer costs $8 per container. Maya needs at least 20 bags of soil for the garden beds. Write an inequality that represents all possible combinations of soil bags (s) and fertilizer containers (f) that Maya can purchase while staying within her budget and meeting the minimum soil requirement. Answer: 4s + 8f ≤ 200, s ≥ 20 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 soil requirement: s ≥ 20 Combine both constraints to get the system: 4s + 8f ≤ 200 and s ≥ 20 The inequality system represents all possible combinations…
   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 soil requirement: s ≥ 20 Step 4: Combine both constraints to get the system: 4s + 8f ≤ 200 and s ≥ 20 Step 5: The inequality system represents all possible combinations of soil and fertilizer Maya can buy.

3. Liam is organizing a school fundraiser and needs to buy snacks. He has a budget of $60 to spend on juice boxes that cost $2 each and granola bars that cost $1.50 each. He needs at least 15 juice boxes and wants to buy at least twice as many granola bars as juice boxes. Write a system of inequalities that represents this situation, using j for juice boxes and g for granola bars. Answer: 2j + 1.5g ≤ 60, j ≥ 15, g ≥ 2j Solution: - \( j \) = number of juice boxes - \( g \) = number of granola bars Juice boxes cost $2 each, so total cost for juice = \( 2j \) dollars. Granola bars cost $1.50 each, so total cost for granola = \( 1.5g \) dollars.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Define the variables** We are told: - \( j \) = number of juice boxes - \( g \) = number of granola bars --- **Step 2: Budget constraint** Juice boxes cost $2 each, so total cost for juice = \( 2j \) dollars. Granola bars cost $1.50 each, so total cost for granola = \( 1.5g \) dollars. Total cost = \( 2j + 1.5g \). Budget is $60, so total cost must be less than or equal to 60: \[ 2j + 1.5g \leq 60 \] --- **Step 3: Minimum juice boxes** Liam needs at least 15 juice boxes: \[ j \geq 15 \] --- **Step 4: Granola bars compared to juice boxes** He wants at least twice as many granola bars as juice boxes: "Twice as many granola bars as juice boxes" means \( g \geq 2j \). --- **Step 5: Write the system** The system of inequalities is: \[ 2j + 1.5g \leq 60 \] \[ j \geq 15 \] \[ g \geq 2j \] --- **Final answer:** 2j + 1.5g ≤ 60, j ≥ 15, g ≥ 2j

4. Maya 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 10 bags of soil and at least 5 bags of fertilizer. Write a system of inequalities that represents the constraints on the number of soil bags (s) and fertilizer bags (f) Maya can purchase while staying within her budget and meeting the minimum requirements. Answer: 8s + 12f ≤ 200, s ≥ 10, f ≥ 5 Solution: Identify the cost inequality. Each soil bag costs $8, so s soil bags cost 8s dollars. Each fertilizer bag costs $12, so f fertilizer bags cost 12f dollars.
   Full step-by-step solution

   Step 1: Identify the cost inequality. Each soil bag costs $8, so s soil bags cost 8s dollars. Each fertilizer bag costs $12, so f fertilizer bags cost 12f dollars. The total cost must be less than or equal to the $200 budget: 8s + 12f ≤ 200 Step 2: Identify the minimum soil requirement. Maya needs at least 10 bags of soil: s ≥ 10 Step 3: Identify the minimum fertilizer requirement. Maya needs at least 5 bags of fertilizer: f ≥ 5 Step 4: Combine all inequalities into a system: 8s + 12f ≤ 200, s ≥ 10, f ≥ 5 The system of inequalities is 8s + 12f ≤ 200, s ≥ 10, f ≥ 5.

5. A rectangular coordinate plane shows the solution region for a system of inequalities. The region is bounded by the lines y = 2x - 4, y = -x + 5, and x = 0. The solution region is shaded where y ≤ 2x - 4, y ≥ -x + 5, and x ≥ 0. What are the coordinates of the vertex point where the lines y = 2x - 4 and y = -x + 5 intersect within this solution region? Answer: (3, 2) Solution: Find the intersection point of y = 2x - 4 and y = -x + 5 by setting them equal: 2x - 4 = -x + 5 Solve for x: 2x + x = 5 + 4 → 3x = 9 → x = 3 Substitute x = 3 into either equation to find y: y = 2(3) - 4 = 6 - 4 = 2 Check if this point satisfies all conditions: For y ≤ 2x - 4: 2 ≤ 2(3) - 4 → 2 ≤…
   Full step-by-step solution

   Step 1: Find the intersection point of y = 2x - 4 and y = -x + 5 by setting them equal: 2x - 4 = -x + 5 Step 2: Solve for x: 2x + x = 5 + 4 → 3x = 9 → x = 3 Step 3: Substitute x = 3 into either equation to find y: y = 2(3) - 4 = 6 - 4 = 2 Step 4: Check if this point satisfies all conditions: For y ≤ 2x - 4: 2 ≤ 2(3) - 4 → 2 ≤ 2 ✓; For y ≥ -x + 5: 2 ≥ -3 + 5 → 2 ≥ 2 ✓; For x ≥ 0: 3 ≥ 0 ✓ Step 5: The intersection point is (3, 2) and it satisfies all inequality conditions. The answer is (3, 2).

6. Is (8, 15) a solution to y > 2x - 7? Answer: No Solution: Substitute x = 8 and y = 15 into the inequality y > 2x - 7. Left side: y = 15. Right side: 2(8) - 7 = 16 - 7 = 9.
   Full step-by-step solution

   Step 1: Substitute x = 8 and y = 15 into the inequality y > 2x - 7. Step 2: Left side: y = 15. Step 3: Right side: 2(8) - 7 = 16 - 7 = 9. Step 4: Check the inequality: 15 > 9 is true. Step 5: Since 15 is greater than 9, the inequality holds true. Therefore, (8, 15) is a solution to y > 2x - 7. The answer is Yes.