- Analyze the absolute value equation \(|x - 72.5| = 4\).
- Solve the two cases: \(x - 72.5 = 4\) and \(x - 72.5 = -4\).
- Calculate the values: \(x = 76.5\) and \(x = 68.5\).
- The range of temperatures is \(\boxed{68.5^\circ\text{F} \text{ and } 76.5^\circ\text{F}}\).
Explanation
- Problem Analysis
The problem asks us to find the range of temperatures in a house given by the absolute value equation \(|x - 72.5| = 4\), where \(x\) represents the temperature in degrees Fahrenheit. The expression \(|x - a| = d\) describes the set of all points \(x\) that are a distance \(d\) from a center point \(a\). In this specific case, the center temperature is \(72.5^\circ\text{F}\) and the allowed variation (distance) from this center is \(4^\circ\text{F}\).
- Setting up the Equations
To solve the absolute value equation \(|x - 72.5| = 4\), we use the property that an equation of the form \(|u| = d\) (where \(d \ge 0\)) is equivalent to two separate equations: \(u = d\) or \(u = -d\). Applying this to our equation, we set up two linear equations:
\[x - 72.5 = 4\]
\[x - 72.5 = -4\]
- Solving the Linear Equations
Now, we solve each of the two linear equations for \(x\):
- For the first equation:
\[x - 72.5 = 4\]
\[x = 72.5 + 4\]
\[x = 76.5\]
- For the second equation:
\[x - 72.5 = -4\]
\[x = 72.5 - 4\]
\[x = 68.5\]
These two solutions represent the maximum and minimum temperatures, respectively.
- Conclusion
Comparing our calculated values, the minimum temperature is \(68.5^\circ\text{F}\) and the maximum temperature is \(76.5^\circ\text{F}\). This corresponds to the second option provided in the problem. The final result is the range of temperatures between \(68.5^\circ\text{F}\) and \(76.5^\circ\text{F}\). We can conclude that the answer is \(\boxed{68.5^\circ\text{F} \text{ and } 76.5^\circ\text{F}}\).
Examples
Absolute value equations are widely used in engineering and home maintenance to define tolerances. For example, if a refrigerator is set to a target temperature of \(38^\circ\text{F}\) with a tolerance of \(2^\circ\text{F}\), the temperature range is governed by \(|x - 38| = 2\). This ensures that the contents of the refrigerator stay within a safe temperature zone, protecting food from spoiling or freezing. By solving these equations, homeowners and technicians can ensure that equipment operates within specified safety and efficiency bounds.