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Difficulty: HardLinear Inequalities in One Variable

The temperature, TT, in degrees Fahrenheit (∘F^\circ\text{F}), of a laboratory incubator mm minutes after a cooling cycle begins is modeled by the equation T=95−0.8(2m−5)T = 95 - 0.8(2m - 5). For a specific experiment, the incubator temperature must be at most 75∘F75^\circ\text{F}. Which of the following inequalities represents all possible values of mm for which the incubator temperature meets this requirement?

  1. m≥15m \geq 15Answer
  2. B
    m≤15m \leq 15
  3. C
    m≥10m \geq 10
  4. D
    m≥30m \geq 30

Answer

The correct inequality is m≥15m \geq 15.
The correct inequality is m≥15m \geq 15. To find this, set the temperature model 95−0.8(2m−5)95 - 0.8(2m - 5) to be less than or equal to 7575. Subtracting 9595 from both sides gives −0.8(2m−5)≤−20-0.8(2m - 5) \leq -20. Dividing both sides by −0.8-0.8 requires flipping the inequality sign, which yields 2m−5≥252m - 5 \geq 25. Adding 55 to both sides gives 2m≥302m \geq 30, and dividing by 22 yields m≥15m \geq 15.

Step-by-Step Solution

1
Set up the inequality based on the requirement that the temperature must be at most 75∘F75^\circ\text{F}.
95−0.8(2m−5)≤7595 - 0.8(2m - 5) \leq 75
The phrase 'at most' corresponds to a less-than-or-equal-to sign.
2
Subtract 9595 from both sides of the inequality.
−0.8(2m−5)≤−20-0.8(2m - 5) \leq -20
This begins the process of isolating the term containing the variable mm.
3
Divide both sides by −0.8-0.8 and flip the inequality sign.
2m−5≥252m - 5 \geq 25
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.
4
Add 55 to both sides of the inequality.
2m≥302m \geq 30
This isolates the variable term 2m2m.
5
Divide both sides by 22 to solve for mm.
m≥15m \geq 15
This yields the solution set for the variable mm.

Key Concept

Solving multi-step linear inequalities in one variable, specifically expanding using the distributive property and reversing the inequality sign when multiplying or dividing by a negative number.
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Topics

Linear Inequalities in One Variable

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