The heat in the house is set to keep the minimum and maximum temperatures (in degrees Fahrenheit) according to the equation [tex]|x-72.5|=4$[/tex]. What are the minimum and maximum temperatures in the house?
A. [tex]$72.5^{\circ} F$[/tex] and [tex]$76.5^{\circ} F$[/tex]
B. [tex]$68.5^{\circ} F$[/tex] and [tex]$76.5^{\circ} F$[/tex]
C. [tex]$70.5^{\circ} F$[/tex] and [tex]$74.5^{\circ} F$[/tex]
D. [tex]$72.5^{\circ} F$[/tex] and [tex]$74.5^{\circ} F$[/tex]
Answer (2)
Split the absolute value equation into two separate equations: x − 72.5 = 4 and x − 72.5 = − 4 .
Solve x − 72.5 = 4 for x to find the maximum temperature: x = 76.5 .
Solve x − 72.5 = − 4 for x to find the minimum temperature: x = 68.5 .
The minimum and maximum temperatures are 68. 5 ∘ F and 76. 5 ∘ F .
Explanation
Understanding the Problem We are given the equation ∣ x − 72.5∣ = 4 , which represents the minimum and maximum temperatures in a house. Our goal is to find these minimum and maximum temperatures.
Splitting the Absolute Value Equation The absolute value equation ∣ x − 72.5∣ = 4 can be split into two separate equations:
x − 72.5 = 4
x − 72.5 = − 4
Solving for the Maximum Temperature Let's solve the first equation, x − 72.5 = 4 , to find the maximum temperature. We add 72.5 to both sides of the equation:
x = 4 + 72.5 = 76.5
So, the maximum temperature is 76. 5 ∘ F .
Solving for the Minimum Temperature Now, let's solve the second equation, x − 72.5 = − 4 , to find the minimum temperature. We add 72.5 to both sides of the equation:
x = − 4 + 72.5 = 68.5
So, the minimum temperature is 68. 5 ∘ F .
Final Answer Therefore, the minimum and maximum temperatures in the house are 68. 5 ∘ F and 76. 5 ∘ F , respectively.
Examples
Absolute value equations are useful in many real-world scenarios. For example, in manufacturing, absolute value equations can be used to specify tolerances. If a machine is supposed to cut a board to be 5 feet long, but it is acceptable for the board to be off by up to 0.1 feet, the length x must satisfy ∣ x − 5∣ ≤ 0.1 . Absolute value equations are also used in physics to calculate errors in measurements.
The minimum temperature in the house is 68. 5 ∘ F and the maximum temperature is 76. 5 ∘ F . So, the answer is option B: 68. 5 ∘ F and 76. 5 ∘ F .
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