Linear Inequalities

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Answer Key & Explanations

Linear Inequalities · Grade 8 · Worksheet 1

1. Is (4, 8) a solution to y ≥ 2x + 2? Answer: No Solution: Substitute x = 4 and y = 8 into the inequality y ≥ 2x + 2. Left side: y = 8. Right side: 2(4) + 2 = 8 + 2 = 10.
   Full step-by-step solution

   Step 1: Substitute x = 4 and y = 8 into the inequality y ≥ 2x + 2. Step 2: Left side: y = 8. Step 3: Right side: 2(4) + 2 = 8 + 2 = 10. Step 4: Check the inequality: 8 ≥ 10 is false because 8 is less than 10. Step 5: Since the inequality is false, the point (4, 8) is not a solution. The answer is No.

2. 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

3. Is (9, 13) a solution to y ≥ 4x - 23? Answer: Yes Solution: Substitute x = 9 and y = 13 into the inequality y ≥ 4x - 23. Left side: y = 13. Right side: 4(9) - 23 = 36 - 23 = 13.
   Full step-by-step solution

   Step 1: Substitute x = 9 and y = 13 into the inequality y ≥ 4x - 23. Step 2: Left side: y = 13. Step 3: Right side: 4(9) - 23 = 36 - 23 = 13. Step 4: Check the inequality: 13 ≥ 13. Step 5: Since 13 is greater than or equal to 13, the inequality is true. Therefore, (9, 13) is a solution to y ≥ 4x - 23. The answer is Yes.

4. Liam is organizing a school fundraiser and needs to buy snacks. He has a budget of $120 to spend on juice boxes that cost $2 each and granola bars that cost $3 each. He needs to buy at least 20 juice boxes and at least 15 granola bars to meet demand. Write a system of inequalities that represents the constraints on the number of juice boxes (j) and granola bars (g) Liam can purchase within his budget while meeting the minimum requirements. Answer: 2j + 3g ≤ 120, j ≥ 20, g ≥ 15 Solution: j = number of juice boxes g = number of granola bars Juice boxes cost $2 each → cost for j juice boxes = 2j dollars Granola bars cost $3 each → cost for g granola bars = 3g dollars Total cost = 2j + 3g Liam's budget is $120, so total cost must be less than or equal to 120: 2j + 3g \leq 120 He…
   Full step-by-step solution

   Let's break down the problem step by step. --- **Step 1: Define the variables** Let j = number of juice boxes g = number of granola bars --- **Step 2: Identify the cost constraint** Juice boxes cost $2 each → cost for j juice boxes = 2j dollars Granola bars cost $3 each → cost for g granola bars = 3g dollars Total cost = 2j + 3g Liam's budget is $120, so total cost must be less than or equal to 120: \[ 2j + 3g \leq 120 \] --- **Step 3: Identify the minimum requirements** He needs at least 20 juice boxes: \[ j \geq 20 \] He needs at least 15 granola bars: \[ g \geq 15 \] --- **Step 4: Combine all constraints** We have: 1. Budget constraint: \( 2j + 3g \leq 120 \) 2. Minimum juice boxes: \( j \geq 20 \) 3. Minimum granola bars: \( g \geq 15 \) Also, j and g are whole numbers (since you can't buy a fraction of a juice box or granola bar), but the problem only asks for the system of inequalities, so we list the three above. --- **Final answer:** 2j + 3g ≤ 120, j ≥ 20, g ≥ 15

5. Is (3, -5) a solution to y > 3x - 7? Answer: No Solution: Substitute x = 3 and y = -5 into y > 3x - 7. Left side: y = -5. Right side: 3(3) - 7 = 9 - 7 = 2.
   Full step-by-step solution

   Step 1: Substitute x = 3 and y = -5 into y > 3x - 7. Step 2: Left side: y = -5. Step 3: Right side: 3(3) - 7 = 9 - 7 = 2. Step 4: Check the inequality: -5 > 2. Step 5: Since -5 is not greater than 2, the statement is false. The point (3, -5) is NOT a solution to y > 3x - 7.

6. Liam is organizing a school fundraiser and needs to buy snacks. He can spend at most $60 on juice boxes and granola bars. Juice boxes cost $3 each and granola bars cost $2 each. He needs at least 5 juice boxes and at least 10 granola bars. Write an inequality system that represents the constraints on the number of juice boxes (j) and granola bars (g) Liam can purchase. Answer: 3j + 2g ≤ 60, j ≥ 5, g ≥ 10 Solution: j = number of juice boxes g = number of granola bars Juice boxes cost $3 each → cost for j juice boxes = 3j dollars Granola bars cost $2 each → cost for g granola bars = 2g dollars Total cost = 3j + 2g Liam can spend at most $60, meaning total cost ≤ 60.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define the variables** Let j = number of juice boxes g = number of granola bars --- **Step 2: Cost constraint** Juice boxes cost $3 each → cost for j juice boxes = 3j dollars Granola bars cost $2 each → cost for g granola bars = 2g dollars Total cost = 3j + 2g Liam can spend at most $60, meaning total cost ≤ 60. So: 3j + 2g ≤ 60 --- **Step 3: Minimum juice boxes** He needs at least 5 juice boxes → j must be greater than or equal to 5. So: j ≥ 5 --- **Step 4: Minimum granola bars** He needs at least 10 granola bars → g must be greater than or equal to 10. So: g ≥ 10 --- **Step 5: Combine all constraints** The system of inequalities is: 3j + 2g ≤ 60 j ≥ 5 g ≥ 10 --- **Final answer:** 3j + 2g ≤ 60, j ≥ 5, g ≥ 10

7. 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).