8

Set 10: Linear Inequalities

Explanation

Answer: A

A manufacturer makes two products, A and B. Product A requires 2 hours of labor, Product B requires 3 hours. Total labor available is 120 hours. They must make at least 10 of Product A. Which system models this?

A.

2A+3B≤120,A≥10,A≥0,B≥02A + 3B \leq 120, A \geq 10, A \geq 0, B \geq 0

✓ Correct
B.

2A+3B≥120,A≥10,A≥0,B≥02A + 3B \geq 120, A \geq 10, A \geq 0, B \geq 0

C.

3A+2B≤120,A≥10,A≥0,B≥03A + 2B \leq 120, A \geq 10, A \geq 0, B \geq 0

D.

2A+3B≤120,B≥10,A≥0,B≥02A + 3B \leq 120, B \geq 10, A \geq 0, B \geq 0

Detailed Explanation

Choice A is correct. Choice A is the correct answer. Translate constraints. 1. Labor: 2A+3B≤1202A + 3 B \leq 120 (Total hours constraint) 2. Quantity: A≥10A \geq 10 (At least 10 of A) 3. Non-negativity: A≥0,B≥0A \geq 0, B \geq 0 (Cannot make negative products) Strategic Tip: Always include non-negativity constraints (x≥0,y≥0x \geq 0, y \geq 0) for real-world production problems. Choice B is incorrect because ≥120\geq 120 implies they must use more than available labor. Choice C is incorrect because it swaps the labor hours per product. Choice D is incorrect because it puts the minimum quantity on B instead of A.

Key Steps:

•

The correct answer is 2A+3B≤120,A≥10,A≥0,B≥02A + 3B \leq 120, A \geq 10, A \geq 0, B \geq 0

Why others are wrong:
B: Choice B is incorrect and may result from a calculation error.
C: Choice C is incorrect and may result from a calculation error.
D: Choice D is incorrect and may result from a calculation error.
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