The heat in the house is set to keep the minimum and maximum temperatures (in degrees Fahrenheit) according to the equation $|x-72.5|=4$. What are the minimum and maximum temperatures in the house?
Answer (1)
Split the absolute value equation ∣ x − 72.5∣ = 4 into two equations: x − 72.5 = 4 and x − 72.5 = − 4 .
Solve x − 72.5 = 4 to find the maximum temperature: x = 76.5 .
Solve x − 72.5 = − 4 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. We need 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 Solving the first equation, x − 72.5 = 4 , we add 72.5 to both sides:
x = 4 + 72.5
x = 76.5
Solving for the Minimum Temperature Solving the second equation, x − 72.5 = − 4 , we add 72.5 to both sides:
x = − 4 + 72.5
x = 68.5
Final Answer Therefore, the minimum temperature is 68. 5 ∘ F and the maximum temperature is 76. 5 ∘ F .
Examples
Absolute value equations are useful in many real-world scenarios, such as determining acceptable ranges in manufacturing or setting tolerances in engineering. For instance, if a machine part needs to be 5 cm long with a tolerance of 0.1 cm, the actual length x must satisfy the equation ∣ x − 5∣ ≤ 0.1 . This ensures the part functions correctly within the specified limits. Similarly, in setting thermostat temperatures, the absolute value equation helps define the acceptable temperature range around a desired setpoint, ensuring comfort and energy efficiency.
