Solve the equation for the given variable:
|x + 8| = 4
Select each correct answer:
The problem requires solving the absolute value equation \(|x + 8| = 4\). The absolute value of a number represents its distance from zero on the number line. Therefore, for an expression \(|u| = k\) where \(k > 0\), the value inside the absolute value, \(u\), must be either \(k\) or \(-k\). In this specific case, \(u = x + 8\) and \(k = 4\), so we need to solve two distinct linear equations to find all possible values of \(x\).
To solve the equation, we split it into two separate cases based on the property of absolute values:
Case 1: \(x + 8 = 4\)
To isolate \(x\), we subtract 8 from both sides of the equation:
\(x = 4 - 8\)
\(x = -4\)
Case 2: \(x + 8 = -4\)
Again, we subtract 8 from both sides to isolate \(x\):
\(x = -4 - 8\)
\(x = -12\)
Thus, the potential solutions are \(x = -4\) and \(x = -12\).
Now that we have determined the two solutions, \(x = -4\) and \(x = -12\), we match them with the provided options:
Both \(-4\) and \(-12\) satisfy the original equation:
Check \(x = -4\): \(|-4 + 8| = |4| = 4\).
Check \(x = -12\): \(|-12 + 8| = |-4| = 4\).
Both checks confirm the solutions are correct. The correct options are (a) and (b).
Absolute value equations are essential when dealing with ranges or tolerances. For example, if a machine is designed to cut a pipe to a length of \(8\) inches with a tolerance of \(4\) inches, the actual length \(x\) of the pipe must satisfy the equation \(|x - 8| \le 4\). In this problem, solving \(|x + 8| = 4\) helps in understanding how to find specific boundary points or values that maintain a constant distance from a target, which is fundamental in engineering, quality control, and statistics.