Linear Inequalities

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Answer Key & Explanations

Linear Inequalities · Grade 8 · Worksheet 3

1. Maria is planning a community garden and needs to buy soil and fertilizer. She has a budget of $200. Soil bags cost $4 each and fertilizer bags cost $10 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 Maria's situation, where s represents the number of soil bags and f represents the number of fertilizer bags. Answer: 4s + 10f ≤ 200, s ≥ 15, f ≥ 5 Solution: Identify the cost constraint. Soil costs $4 per bag, fertilizer costs $10 per bag, and the total budget is $200. Maria needs at least 15 bags of soil, so: s ≥ 15 Identify the minimum fertilizer requirement.
   Full step-by-step solution

   Step 1: Identify the cost constraint. Soil costs $4 per bag, fertilizer costs $10 per bag, and the total budget is $200. So the cost inequality is: 4s + 10f ≤ 200 Step 2: Identify the minimum soil requirement. Maria needs at least 15 bags of soil, so: s ≥ 15 Step 3: Identify the minimum fertilizer requirement. Maria wants at least 5 bags of fertilizer, so: f ≥ 5 Step 4: Combine all constraints into a system of inequalities: 4s + 10f ≤ 200 s ≥ 15 f ≥ 5 The complete system is: 4s + 10f ≤ 200, s ≥ 15, f ≥ 5

2. Emma 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 $10 per bag. She needs at least 20 bags of soil for the garden beds and wants to buy 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) Emma can purchase while staying within her budget and meeting her requirements. Answer: s ≥ 20, f ≥ 5, 4s + 10f ≤ 200 Solution: Budget limitations create upper bounds on spending, while minimum requirements create lower bounds on quantities.
   Full step-by-step solution

   In real-world planning situations, we often have multiple constraints to consider simultaneously. Budget limitations create upper bounds on spending, while minimum requirements create lower bounds on quantities. These constraints can be represented mathematically using inequalities, where each inequality captures one aspect of the overall situation.

3. Is (7, 22) a solution to y ≥ 3x + 2? Answer: No Solution: Substitute x = 7 and y = 22 into the inequality y ≥ 3x + 2. Left side: y = 22. Right side: 3(7) + 2 = 21 + 2 = 23.
   Full step-by-step solution

   Step 1: Substitute x = 7 and y = 22 into the inequality y ≥ 3x + 2. Step 2: Left side: y = 22. Step 3: Right side: 3(7) + 2 = 21 + 2 = 23. Step 4: Check the inequality: 22 ≥ 23 is false because 22 is less than 23. Step 5: Since the inequality is false, the point (7, 22) is not a solution. The answer is No.

4. Mere is organizing a school cultural festival and needs to buy two types of fabric for costumes: cotton fabric costs $14 per meter and silk fabric costs $22 per meter. She has a total budget of $440. She needs at least 12 meters of silk fabric for the main costumes. Let x represent the number of meters of cotton fabric and y represent the number of meters of silk fabric. Write a system of inequalities to represent this situation, then determine if Mere can buy 15 meters of cotton fabric and 14 meters of silk fabric while staying within her budget and meeting the silk requirement. Answer: No Solution: Write the cost inequality. Cotton costs $14 per meter and silk costs $22 per meter, so total cost is 14x + 22y. The budget is $440, so: 14x + 22y ≤ 440.
   Full step-by-step solution

   Step 1: Write the cost inequality. Cotton costs $14 per meter and silk costs $22 per meter, so total cost is 14x + 22y. The budget is $440, so: 14x + 22y ≤ 440. Step 2: Write the minimum silk requirement. She needs at least 12 meters of silk, so: y ≥ 12. Step 3: Test the point (15, 14) in the cost inequality: 14(15) + 22(14) = 210 + 308 = 518. Since 518 > 440, the cost inequality is not satisfied. Step 4: Test the point (15, 14) in the silk requirement: 14 ≥ 12, so this inequality is satisfied. Step 5: Since both inequalities must be true, and the cost inequality fails, Mere cannot buy this combination. Final answer: No

5. Which point satisfies both inequalities: 3x - 2y < 6 and x + y ≥ 4? Answer: C. (3, 2) Solution: Check point A (1, 2) 3(1) - 2(2) = 3 - 4 = -1 < 6 ✓ 1 + 2 = 3 ≥ 4 ✗ (3 is less than 4) Check point B (2, 1) 3(2) - 2(1) = 6 - 2 = 4 < 6 ✓ 2 + 1 = 3 ≥ 4 ✗ (3 is less than 4) Check point C (3, 2) 3(3) - 2(2) = 9 - 4 = 5 < 6 ✓ 3 + 2 = 5 ≥ 4 ✓ Check point D (0, 4) 3(0) - 2(4) = 0 - 8 = -8 < 6 ✓ 0 +…
   Full step-by-step solution

   Step 1: Check point A (1, 2) 3(1) - 2(2) = 3 - 4 = -1 < 6 ✓ 1 + 2 = 3 ≥ 4 ✗ (3 is less than 4) Point A fails the second inequality Step 2: Check point B (2, 1) 3(2) - 2(1) = 6 - 2 = 4 < 6 ✓ 2 + 1 = 3 ≥ 4 ✗ (3 is less than 4) Point B fails the second inequality Step 3: Check point C (3, 2) 3(3) - 2(2) = 9 - 4 = 5 < 6 ✓ 3 + 2 = 5 ≥ 4 ✓ Point C satisfies both inequalities Step 4: Check point D (0, 4) 3(0) - 2(4) = 0 - 8 = -8 < 6 ✓ 0 + 4 = 4 ≥ 4 ✓ Point D satisfies both inequalities Step 5: Both points C and D satisfy both inequalities, but the question asks for which point satisfies both, and C is one correct answer. The correct answer is (3, 2).

6. A rectangular coordinate plane shows the solution region for a system of inequalities. The first inequality is y > 2x - 3. The second inequality is y ≤ -x + 4. The third inequality is x ≥ 0. Which of the following points lies within the solution region where all three inequalities are satisfied? Answer: (2,1) Solution: Test point (0,0) in all three inequalities: For y > 2x - 3: 0 > 2(0) - 3 → 0 > -3 → TRUE For y ≤ -x + 4: 0 ≤ -0 + 4 → 0 ≤ 4 → TRUE For x ≥ 0: 0 ≥ 0 → TRUE Point (0,0) satisfies all three inequalities.
   Full step-by-step solution

   Step 1: Test point (0,0) in all three inequalities: For y > 2x - 3: 0 > 2(0) - 3 → 0 > -3 → TRUE For y ≤ -x + 4: 0 ≤ -0 + 4 → 0 ≤ 4 → TRUE For x ≥ 0: 0 ≥ 0 → TRUE Point (0,0) satisfies all three inequalities. Step 2: Test point (3,4) in all three inequalities: For y > 2x - 3: 4 > 2(3) - 3 → 4 > 6 - 3 → 4 > 3 → TRUE For y ≤ -x + 4: 4 ≤ -3 + 4 → 4 ≤ 1 → FALSE Point (3,4) does NOT satisfy all three inequalities. Step 3: Test point (2,1) in all three inequalities: For y > 2x - 3: 1 > 2(2) - 3 → 1 > 4 - 3 → 1 > 1 → FALSE Point (2,1) does NOT satisfy all three inequalities. Step 4: Test point (1,3) in all three inequalities: For y > 2x - 3: 3 > 2(1) - 3 → 3 > 2 - 3 → 3 > -1 → TRUE For y ≤ -x + 4: 3 ≤ -1 + 4 → 3 ≤ 3 → TRUE For x ≥ 0: 1 ≥ 0 → TRUE Point (1,3) satisfies all three inequalities. Step 5: Test point (4,-2) in all three inequalities: For y > 2x - 3: -2 > 2(4) - 3 → -2 > 8 - 3 → -2 > 5 → FALSE Point (4,-2) does NOT satisfy all three inequalities. After testing all points, only (0,0) and (1,3) satisfy all three inequalities. The problem asks for a point that lies within the solution region, and since (0,0) lies exactly on the boundary line x=0, the point that lies within the region is (1,3).

7. Is (4, 6) a solution to y ≥ 2x - 4? Answer: Yes Solution: Substitute x = 4 and y = 6 into the inequality y ≥ 2x - 4. Left side: y = 6. Right side: 2(4) - 4 = 8 - 4 = 4.
   Full step-by-step solution

   Step 1: Substitute x = 4 and y = 6 into the inequality y ≥ 2x - 4. Step 2: Left side: y = 6. Step 3: Right side: 2(4) - 4 = 8 - 4 = 4. Step 4: Check the inequality: 6 ≥ 4. Step 5: Since 6 is greater than or equal to 4, the inequality is true. Therefore, (4, 6) is a solution to y ≥ 2x - 4. The answer is Yes.