Select the correct answer.
What is the solution to the equation $|5-x|=13$ ?
To solve the equation \(|5 - x| = 13\), we first recall the definition of an absolute value equation. For any expression \(u\), the equation \(|u| = k\) (where \(k > 0\)) is equivalent to two separate cases: \(u = k\) or \(u = -k\). In this problem, our expression is \(u = 5 - x\) and our constant is \(k = 13\). Therefore, the equation splits into two linear equations.
The first case is \(5 - x = 13\). To solve for \(x\), we subtract 5 from both sides:
\(-x = 13 - 5\)
\(-x = 8\)
Multiplying by -1, we find:
\(x = -8\)
The second case is \(5 - x = -13\). To solve for \(x\), we subtract 5 from both sides:
\(-x = -13 - 5\)
\(-x = -18\)
Multiplying by -1, we find:
\(x = 18\)
By solving the absolute value equation, we have identified two possible values for \(x\): \(x = -8\) and \(x = 18\). Looking at the provided options, the correct choice is \(x = -8\) or \(x = 18\).
The concept of absolute value is essential in measuring distances or differences, as these values are always non-negative. For instance, if you are measuring the temperature deviation from a set point (e.g., 5 degrees) in a laboratory experiment, the absolute value helps you quantify how far the current temperature is from the target, regardless of whether it is warmer or colder. Understanding this allows you to determine the range of acceptable temperatures needed for a process to remain stable.