GMAT Practice Question: 7x + 6y \le 38,000 4x + 5y \le 28,000 A manufacturer wants to produce x balls and y...
Question
7x + 6y \le 38,000 4x + 5y \le 28,000 A manufacturer wants to produce x balls and y boxes. Resource constraints require that x and y satisfy the inequalities shown. What is the maximum number of balls and boxes combined that can be produced given the resource constraints?
- 5,000
- 6,000
- 7,000
- 8,000
- 10,000
Topics: inequalities
Solution
Step 1
We first name the variables to match the question.
x = \text{number of balls} y = \text{number of boxes}
Step 2
We sum the two resource constraints \textbf{"7x + 6y \leq 38000"} and \textbf{"4x + 5y \leq 28000"}, applying \textbf{Adding Two Inequalities}.
7x + 6y \le 38000 4x + 5y \le 28000 11x + 11y \le 66000
Step 3
We divide both sides of the combined inequality by 11, which is positive, so the inequality direction stays the same according to \textbf{Multiplying/Dividing Inequalities by Positive}.
\frac{11x + 11y}{11} \le \frac{66000}{11} x + y \le 6000
Step 4
We verify that the bound is attainable by finding values that satisfy both original constraints.
7 \times 2000 + 6 \times 4000 = 38000 4 \times 2000 + 5 \times 4000 = 28000 x + y = 2000 + 4000 = 6000
Answer
6000
