Linear Inequalities

lessonbunny.com

Answer Key & Explanations

Linear Inequalities · Grade 8 · Worksheet 2

1. Is (11, 9) a solution to y ≤ 2x - 13? Answer: No Solution: Substitute x = 11 and y = 9 into the inequality y ≤ 2x - 13. Left side: y = 9. Right side: 2(11) - 13 = 22 - 13 = 9.
   Full step-by-step solution

   Step 1: Substitute x = 11 and y = 9 into the inequality y ≤ 2x - 13. Step 2: Left side: y = 9. Step 3: Right side: 2(11) - 13 = 22 - 13 = 9. Step 4: Check the inequality: 9 ≤ 9 is true because 9 is equal to 9. Step 5: Since the inequality is true, the point (11, 9) is a solution. The answer is Yes.

2. A company is manufacturing two types of solar panels. Model X requires 3 hours of assembly time and 2 hours of testing time. Model Y requires 4 hours of assembly time and 1 hour of testing time. The factory has at most 240 hours of assembly time available per week and at most 100 hours of testing time available per week. If x represents the number of Model X panels and y represents the number of Model Y panels produced per week, which system of inequalities represents these constraints? Answer: 3x + 4y ≤ 240, 2x + y ≤ 100 Solution: - Each Model X requires 3 hours of assembly: 3x - Each Model Y requires 4 hours of assembly: 4y - Total assembly time used: 3x + 4y - Assembly time available is at most 240 hours: 3x + 4y ≤ 240 - Each Model X requires 2 hours of testing: 2x - Each Model Y requires 1 hour of testing: 1y - Total…
   Full step-by-step solution

   Step 1: Analyze the assembly time constraint - Each Model X requires 3 hours of assembly: 3x - Each Model Y requires 4 hours of assembly: 4y - Total assembly time used: 3x + 4y - Assembly time available is at most 240 hours: 3x + 4y ≤ 240 Step 2: Analyze the testing time constraint - Each Model X requires 2 hours of testing: 2x - Each Model Y requires 1 hour of testing: 1y - Total testing time used: 2x + y - Testing time available is at most 100 hours: 2x + y ≤ 100 Step 3: Combine the constraints The complete system of inequalities is: 3x + 4y ≤ 240 2x + y ≤ 100 These inequalities represent the assembly and testing time constraints for producing x Model X panels and y Model Y panels.

3. Maya is planning a community garden and needs to buy fencing for two rectangular plots. The first plot requires at least 3 times as much fencing as the second plot. The total amount of fencing available is 120 feet. If x represents the fencing for the first plot and y represents the fencing for the second plot, which system of inequalities represents this situation? Answer: B. x ≥ 3y, x + y ≤ 120, x ≥ 0, y ≥ 0 Solution: The first plot requires at least 3 times as much fencing as the second plot, so x ≥ 3y The total fencing available is 120 feet, so x + y ≤ 120 Since fencing amounts cannot be negative, we also need x ≥ 0 and y ≥ 0 Putting all constraints together: x ≥ 3y, x + y ≤ 120, x ≥ 0, y ≥ 0 Comparing with…
   Full step-by-step solution

   Step 1: The first plot requires at least 3 times as much fencing as the second plot, so x ≥ 3y Step 2: The total fencing available is 120 feet, so x + y ≤ 120 Step 3: Since fencing amounts cannot be negative, we also need x ≥ 0 and y ≥ 0 Step 4: Putting all constraints together: x ≥ 3y, x + y ≤ 120, x ≥ 0, y ≥ 0 Step 5: Comparing with the options, option C matches all these constraints

4. Aroha is organizing a school craft fair and needs to buy two types of yarn for student projects. Cotton yarn costs $9 per skein, and wool yarn costs $12 per skein. She has a budget of $180 and needs at least 10 skeins of cotton yarn for the basic projects. Write an inequality that represents all possible combinations of cotton skeins (x) and wool skeins (y) Aroha can purchase while staying within her budget and meeting the minimum cotton yarn requirement. Then, determine if she can buy 12 skeins of cotton yarn and 8 skeins of wool yarn. Answer: No Solution: Write the cost inequality. Cotton costs $9 per skein, wool costs $12 per skein, and the budget is $180. So 9x + 12y ≤ 180.
   Full step-by-step solution

   Step 1: Write the cost inequality. Cotton costs $9 per skein, wool costs $12 per skein, and the budget is $180. So 9x + 12y ≤ 180. Step 2: Write the minimum cotton requirement: x ≥ 10. Step 3: Test x = 12 and y = 8 in the cost inequality: 9(12) + 12(8) = 108 + 96 = 204. Step 4: Compare to budget: 204 > 180, so the cost exceeds the budget. Step 5: Check the minimum requirement: 12 ≥ 10, which is satisfied. Step 6: Since the cost inequality is not satisfied, Aroha cannot buy this combination. Final answer: No.

5. 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

6. Is (5, 0) a solution to y ≤ 2x - 5? Answer: Yes Solution: Identify the coordinates. x = 5, y = 0. Substitute into the inequality y ≤ 2x - 5.
   Full step-by-step solution

   Step 1: Identify the coordinates. x = 5, y = 0. Step 2: Substitute into the inequality y ≤ 2x - 5. 0 ≤ 2(5) - 5 Step 3: Simplify the right side. 0 ≤ 10 - 5 0 ≤ 5 Step 4: Check if the statement is true. 0 is less than or equal to 5, so the statement is true. The point (5, 0) satisfies the inequality. The answer is Yes.