What are the solution(s) to the following equation? Select each correct answer.
$|2x - 5| = 11$
The equation provided is \(|2x - 5| = 11\). The absolute value function \(|u|\) measures the distance of \(u\) from zero. An equation of the form \(|u| = a\) (where \(a > 0\)) means that the expression \(u\) is at a distance of \(a\) from zero, which implies that \(u\) must be equal to \(a\) or \(u\) must be equal to \(-a\). In our case, \(u = 2x - 5\) and \(a = 11\), so we need to solve for two separate cases.
Based on the property of absolute values, we split the equation \(|2x - 5| = 11\) into two distinct linear equations:
We solve each case individually.
Case 1:
\(2x - 5 = 11\)
Adding 5 to both sides:
\(2x = 16\)
Dividing by 2:
\(x = 8\)
Case 2:
\(2x - 5 = -11\)
Adding 5 to both sides:
\(2x = -6\)
Dividing by 2:
\(x = -3\)
The solutions to the equation \(|2x - 5| = 11\) are \(x = 8\) and \(x = -3\). Comparing these results with the provided options, we select:
Absolute value equations are essential in many practical scenarios where you are interested in the difference from a target value regardless of the direction. For instance, if a manufacturing machine is supposed to cut metal rods to a target length of 5 cm, but has a tolerance of 11 cm, you would express the range of acceptable variations as \(|L - 5| \le 11\), where \(L\) is the actual length. Solving such equations helps technicians define the boundaries of acceptable quality control in production environments.