Examkin

Question

Difficulty: MediumSolving Linear Inequalities

A laboratory chamber's initial temperature is 72\frac{7}{2} degrees Celsius, and it decreases at a constant rate of 54\frac{5}{4} degrees Celsius per hour. The temperature of the chamber must reach at most −12-\frac{1}{2} degrees Celsius to complete an experiment. Which of the following inequalities represents the number of hours, hh, the experiment must run to reach this temperature?

  1. A
    h≤165h \leq \frac{16}{5}
  2. B
    h≥85h \geq \frac{8}{5}
  3. h≥165h \geq \frac{16}{5}Answer
  4. D
    h≤−29h \leq -\frac{2}{9}
  5. E
    h≥12h \geq \frac{1}{2}

Answer

h≥165h \geq \frac{16}{5}
The correct answer is the inequality stating that hh must be greater than or equal to sixteen-fifths. This is found by setting up the linear inequality 72−54h≤−12\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}, subtracting 72\frac{7}{2} from both sides to get −54h≤−4-\frac{5}{4}h \leq -4, and then multiplying by −45-\frac{4}{5} while reversing the inequality sign.

Step-by-Step Solution

1
Set up the inequality representing the temperature constraint.
72−54h≤−12\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}
The initial temperature is 72\frac{7}{2}, the rate of decrease is 54\frac{5}{4} per hour hh, and the final temperature must be at most −12-\frac{1}{2}.
2
Subtract 72\frac{7}{2} from both sides of the inequality.
−54h≤−4-\frac{5}{4}h \leq -4
This isolates the variable term on the left side of the inequality. The subtraction is −12−72=−82=−4-\frac{1}{2} - \frac{7}{2} = -\frac{8}{2} = -4.
3
Multiply both sides by −45-\frac{4}{5} and reverse the inequality sign.
h≥165h \geq \frac{16}{5}
Multiplying or dividing by a negative number requires reversing the direction of the inequality sign. The calculation is −4×(−45)=165-4 \times \left(-\frac{4}{5}\right) = \frac{16}{5}.

Key Concept

Solving linear inequalities involving negative coefficients and applying the inequality sign-flip rule.
Estimated Time:1m 30s
Rate this question

Topics

Solving Linear Inequalities

More questions from this topic