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Difficulty: HardSolving Linear Inequalities

A manufacturer calculates the net monthly operating profit PP (in thousands of dollars) using the expression 8−3x2\frac{8 - 3x}{2}, where xx is the number of maintenance hours scheduled. The auxiliary support costs are modeled by the expression 2x−13\frac{2x - 1}{3} thousand dollars. For the plant to be considered efficient, the net operating profit minus the auxiliary support costs must be at least 33 thousand dollars. Which of the following inequalities represents the range of maintenance hours, xx, that satisfy this efficiency requirement?

  1. A
    x≤413x \leq \frac{4}{13}
  2. B
    x≥813x \geq \frac{8}{13}
  3. x≤813x \leq \frac{8}{13}Answer
  4. D
    x≤45x \leq \frac{4}{5}
  5. E
    x≥125x \geq \frac{12}{5}

Answer

The range of maintenance hours that satisfy the efficiency requirement is x≤813x \leq \frac{8}{13}.
First, translate the word problem into a mathematical inequality. The net monthly operating profit minus the auxiliary support costs must be at least 3, which translates to: 8−3x2−2x−13≥3\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3 To solve this inequality, find the least common denominator (LCD) of 2 and 3, which is 6. Multiply every term in the inequality by 6: 3(8−3x)−2(2x−1)≥183(8 - 3x) - 2(2x - 1) \geq 18 Distribute the constants on the left side, paying careful attention to distribute the negative sign: 24−9x−4x+2≥1824 - 9x - 4x + 2 \geq 18 Combine like terms: 26−13x≥1826 - 13x \geq 18 Subtract 26 from both sides: −13x≥−8-13x \geq -8 Divide both sides by -13. Because we are dividing by a negative number, the direction of the inequality sign must be reversed: x≤813x \leq \frac{8}{13} Thus, the option stating x≤813x \leq \frac{8}{13} is correct.

Step-by-Step Solution

1
Translate the word problem into a mathematical inequality.
8−3x2−2x−13≥3\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3
The net operating profit minus the auxiliary support costs must be at least 3, which translates to a subtraction operation set greater than or equal to 3.
2
Multiply all terms in the inequality by the least common denominator (LCD) of 2 and 3, which is 6.
3(8−3x)−2(2x−1)≥183(8 - 3x) - 2(2x - 1) \geq 18
Multiplying by the LCD eliminates the denominators and simplifies the equation for algebraic manipulation.
3
Distribute the constants on the left side of the inequality.
24−9x−4x+2≥1824 - 9x - 4x + 2 \geq 18
Removing the parentheses is required to combine like terms. Be careful to distribute the negative sign: −2×−1=+2-2 \times -1 = +2.
4
Combine like terms on the left side of the inequality.
26−13x≥1826 - 13x \geq 18
Simplifying the left-hand side reduces the inequality to a standard two-step linear inequality.
5
Subtract 26 from both sides of the inequality to isolate the variable term.
−13x≥−8-13x \geq -8
This isolates the variable term −13x-13x on the left side.
6
Divide both sides by -13 and reverse the direction of the inequality sign.
x≤813x \leq \frac{8}{13}
Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain equivalence.

Key Concept

Solving multi-step linear inequalities, including fraction clearance and reversing the inequality sign when dividing by a negative number.
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Topics

Solving Linear Inequalities

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