Linear Inequalities

lessonbunny.com

Answer Key & Explanations

Linear Inequalities · Grade 8 · Worksheet 3

1. Isabella is helping her grandmother create two types of gift baskets for a charity auction. Each small basket requires 2 apples and costs $3 to make. Each large basket requires 7 apples and costs $12 to make. Isabella's grandmother has 77 apples available and a budget of $132. Write a system of inequalities that represents the number of small baskets (x) and large baskets (y) they can make, given the constraints. Then, determine if they can make 12 small baskets and 7 large baskets. Answer: No, they cannot make 12 small baskets and 7 large baskets because it exceeds the apple constraint (12*2 + 7*7 = 73 apples used, which is within 77, but the cost is 12*3 + 7*12 = $120, which is within $132; however, the apple constraint is 2x + 7y ≤ 77. For x=12, y=7: 2(12) + 7(7) = 24 + 49 = 73 ≤ 77. Cost: 3x + 12y ≤ 132: 3(12) + 12(7) = 36 + 84 = 120 ≤ 132. Both constraints are satisfied, so the answer is yes.) Solution: Define variables: let x = number of small baskets, y = number of large baskets. Apple constraint: Each small basket uses 2 apples, each large basket uses 7 apples, total apples ≤ 77. So, 2x + 7y ≤ 77.
   Full step-by-step solution

   Step 1: Define variables: let x = number of small baskets, y = number of large baskets. Step 2: Apple constraint: Each small basket uses 2 apples, each large basket uses 7 apples, total apples ≤ 77. So, 2x + 7y ≤ 77. Step 3: Cost constraint: Each small basket costs $3, each large basket costs $12, total cost ≤ $132. So, 3x + 12y ≤ 132. Step 4: Check x = 12 and y = 7 for apple constraint: 2(12) + 7(7) = 24 + 49 = 73. Since 73 ≤ 77, this constraint is satisfied. Step 5: Check cost constraint: 3(12) + 12(7) = 36 + 84 = 120. Since 120 ≤ 132, this constraint is satisfied. Step 6: Both constraints are satisfied, so Isabella and her grandmother can make 12 small baskets and 7 large baskets. The answer is yes.

2. 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.

3. Kaia is organizing a school cultural festival and needs to buy two types of traditional fabric for decorations. She has a budget of $240. Fabric type A costs $9 per meter, and fabric type B costs $12 per meter. Kaia needs at least 12 meters of fabric type B for the main stage backdrop. She also wants to buy at least 3 times as many meters of fabric type A as fabric type B. Write a system of inequalities that represents Kaia's constraints, where x represents the meters of fabric type A and y represents the meters of fabric type B. Then, test whether Kaia can buy 30 meters of fabric type A and 10 meters of fabric type B while meeting all her requirements. Answer: No Solution: Write the budget inequality. Fabric A costs $9 per meter and fabric B costs $12 per meter. The total cost is 9x + 12y.
   Full step-by-step solution

   Step 1: Write the budget inequality. Fabric A costs $9 per meter and fabric B costs $12 per meter. The total cost is 9x + 12y. Since the budget is $240, we write: 9x + 12y ≤ 240. Step 2: Write the minimum requirement for fabric B. Kaia needs at least 12 meters of fabric B, so: y ≥ 12. Step 3: Write the relationship between the two fabrics. She wants at least 3 times as many meters of fabric A as fabric B, so: x ≥ 3y. Step 4: Test the point (x=30, y=10). Check budget: 9(30) + 12(10) = 270 + 120 = 390. This is greater than 240, so the budget inequality is NOT satisfied. Step 5: Check minimum fabric B: y=10 is less than 12, so y ≥ 12 is NOT satisfied. Step 6: Check relationship: x=30, 3y=30, so 30 ≥ 30 is satisfied. Since two of the three conditions fail, Kaia cannot buy 30 meters of fabric A and 10 meters of fabric B. Final answer: No.

4. 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.

5. 2x + 3y ≥ 12; x - y < 4; Find the solution region for x > 0 and y > 0 Answer: The solution region is the area above the line 2x + 3y = 12 and below the line x - y = 4, in the first quadrant Solution: When solving systems of linear inequalities, each inequality divides the coordinate plane into two regions. The solution to the system is the intersection of all these regions.
   Full step-by-step solution

   When solving systems of linear inequalities, each inequality divides the coordinate plane into two regions. The solution to the system is the intersection of all these regions. Remember to test points to verify which side of each boundary line satisfies the inequality