Select the correct answer.
What is the solution to the equation $|5-x|=13$ ?
The absolute value of a number \(z\), denoted by \(|z|\), represents its distance from zero on the number line. When we are given an equation of the form \(|f(x)| = a\), where \(a \geq 0\), it indicates that the expression \(f(x)\) must be at a distance of \(a\) from zero. This means that either \(f(x) = a\) or \(f(x) = -a\). In our specific case, the equation is \(|5 - x| = 13\). This translates to two distinct linear scenarios based on the distance definition: the expression \(5 - x\) must equal either \(13\) or \(-13\).
To find the values of \(x\), we split the absolute value equation into two linear equations:
1) \(5 - x = 13\)
2) \(5 - x = -13\)
For the first equation, \(5 - x = 13\):
Subtract \(5\) from both sides:
\(-x = 13 - 5\)
\(-x = 8\)
Multiply by \(-1\):
\(x = -8\)
For the second equation, \(5 - x = -13\):
Subtract \(5\) from both sides:
\(-x = -13 - 5\)
\(-x = -18\)
Multiply by \(-1\):
\(x = 18\)
Having solved both cases, we have determined that the values of \(x\) that satisfy the original equation are \(x = -8\) and \(x = 18\). Comparing these results to the given options, we identify that the third option, \(x = -8\) or \(x = 18\), is the correct solution.
Absolute value equations are essential in real-world scenarios where you need to measure deviations from a target value regardless of the direction. For instance, if a machine is designed to cut metal rods to a target length of 5 cm, but has an allowed error tolerance of 13 cm, you might solve \(|x - 5| = 13\) to find the minimum and maximum possible lengths produced by the machine. In this case, solving the equation tells us the rods could be as short as \(-8\) cm (which would be physically impossible here, but the math holds) or as long as \(18\) cm. Such calculations are fundamental in quality control, navigation, and statistical analysis.