Solve for x:
$|x-5|=8$
$|x-5|=8$
Solve for x:
$|x-5|=8$
We are given the absolute value equation \(|x - 5| = 8\). The absolute value function \(|a|\) represents the distance of \(a\) from \(0\) on the number line. Consequently, the expression \(|x - 5| = 8\) means that the distance between \(x\) and \(5\) is exactly \(8\). According to the definition of absolute value, for any real number \(a\) and a non-negative constant \(k\), the equation \(|a| = k\) is equivalent to the two cases: \(a = k\) or \(a = -k\). In this problem, \(a = x - 5\) and \(k = 8\).
Based on the property of absolute value equations, we can split the equation \(|x - 5| = 8\) into two separate linear equations:
These two equations represent the two points on the number line that are at a distance of 8 from 5.
Now, we solve each linear equation for \(x\):
Case 1:
\[x - 5 = 8\]
Adding 5 to both sides gives:
\[x = 8 + 5\]
\[x = 13\]
Case 2:
\[x - 5 = -8\]
Adding 5 to both sides gives:
\[x = -8 + 5\]
\[x = -3\]
Therefore, the values of \(x\) that satisfy the original equation are \(13\) and \(-3\).
The concept of absolute value equations like \(|x - 5| = 8\) is useful in real-world scenarios where we care about the distance from a target value, regardless of the direction. For instance, imagine a quality control process for a machine producing metal rods of a specific length, say 5 cm. If the tolerance level is 8 cm, the machine will accept any rod with length \(x\) such that \(|x - 5| = 8\). This means the rods would be either 13 cm or -3 cm (though length must be positive, this model illustrates how absolute value bounds help identify acceptable limits in engineering and manufacturing).