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Difficulty: MediumTranslating and Solving Algebraic Word Problems

A company's net profit is calculated by subtracting its monthly operating costs from its monthly revenue. In a certain month, the company's revenue is 260,000260,000 dollars minus twice its monthly operating costs. If the company wants its monthly net profit to be at least 110,000110,000 dollars, which of the following inequalities represents all possible values of its monthly operating costs, CC, in dollars?

  1. A
    C≥50,000C \ge 50,000
  2. C≤50,000C \le 50,000Answer
  3. C
    C≤75,000C \le 75,000
  4. D
    C≥75,000C \ge 75,000
  5. E
    C≤150,000C \le 150,000

Answer

The inequality C≤50,000C \le 50,000
The correct inequality is obtained by setting up the net profit as the revenue (260,000−2C260,000 - 2C) minus the operating costs (CC), which simplifies to 260,000−3C260,000 - 3C. Setting this profit to be at least 110,000110,000 gives the inequality 260,000−3C≥110,000260,000 - 3C \ge 110,000. Subtracting 260,000260,000 from both sides results in −3C≥−150,000-3C \ge -150,000. Finally, dividing both sides by −3-3 and reversing the inequality sign because of the division by a negative number yields C≤50,000C \le 50,000.

Step-by-Step Solution

1
Define the expression for net profit by subtracting the operating costs, CC, from the revenue, which is 260,000−2C260,000 - 2C.
Net profit = (260,000−2C)−C=260,000−3C(260,000 - 2C) - C = 260,000 - 3C
Net profit is defined as revenue minus operating costs.
2
Set up the inequality where the net profit is at least 110,000110,000.
260,000−3C≥110,000260,000 - 3C \ge 110,000
The phrase 'at least' translates to the 'greater than or equal to' inequality sign (≥\ge).
3
Subtract 260,000260,000 from both sides of the inequality.
−3C≥−150,000-3C \ge -150,000
Isolate the term containing the variable CC.
4
Divide both sides by −3-3 and reverse the inequality sign.
C≤50,000C \le 50,000
Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Translating and Solving Algebraic Word Problems
Estimated Time:1m 30s
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Topics

Translating and Solving Algebraic Word Problems

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