Hiroto solved the equation $6-4|2 x-8|=-10$ for one solution. His work is shown below.
$ \begin{array}{l}
6-4|2 x-8|=-10 \\
-4|2 x-8|=-16 \\
|2 x-8|=4 \\
2 x-8=4 \\
2 x=12 \\
x=6
\end{array} $
What is the other solution?
A. -6
B. -4
C. 2
D. 10
Answer (1)
Isolate the absolute value term: ∣2 x − 8∣ = 4 .
Split the absolute value equation into two cases: 2 x − 8 = 4 and 2 x − 8 = − 4 .
Solve the first case: x = 6 .
Solve the second case: x = 2 . The other solution is 2 .
Explanation
Understanding the Problem We are given the equation 6 − 4∣2 x − 8∣ = − 10 and Hiroto's solution x = 6 . We need to find the other solution.
Isolating the Absolute Value First, isolate the absolute value term by subtracting 6 from both sides of the equation: 6 − 4∣2 x − 8∣ − 6 = − 10 − 6 − 4∣2 x − 8∣ = − 16
Simplifying the Equation Next, divide both sides by -4: − 4 − 4∣2 x − 8∣ = − 4 − 16 ∣2 x − 8∣ = 4
Considering Two Cases Now, we consider two cases for the absolute value: Case 1: 2 x − 8 = 4 Case 2: 2 x − 8 = − 4
Solving Case 1 Case 1: 2 x − 8 = 4 . Add 8 to both sides: 2 x − 8 + 8 = 4 + 8 2 x = 12 Divide by 2: 2 2 x = 2 12 x = 6 This is the solution Hiroto found.
Solving Case 2 Case 2: 2 x − 8 = − 4 . Add 8 to both sides: 2 x − 8 + 8 = − 4 + 8 2 x = 4 Divide by 2: 2 2 x = 2 4 x = 2
Finding the Other Solution Therefore, the other solution is x = 2 .
Examples
Absolute value equations are useful in many real-world scenarios, such as determining tolerances in manufacturing. For example, if a machine is designed to produce parts that are 5 cm long, but a tolerance of 0.1 cm is allowed, the actual length x of the part must satisfy the equation ∣ x − 5∣ l e 0.1 . This means the length can be between 4.9 cm and 5.1 cm. Understanding how to solve absolute value equations helps engineers ensure that manufactured parts meet the required specifications.
