Question

given the function $f(x) = 0.5|x - 4| - 3$, for what values of $x$ is $f(x) = 7$?
$x = -24, x = 16$
$x = -16, x = 24$
$x = -1, x = 9$
$x = 1, x = -9$

Explanation:

Step1: Set up the equation

We know that \( f(x) = 0.5|x - 4| - 3 \) and we want to find \( x \) when \( f(x)=7 \). So we set up the equation:
\( 0.5|x - 4| - 3 = 7 \)

Step2: Isolate the absolute value term

First, add 3 to both sides of the equation:
\( 0.5|x - 4|=7 + 3 \)
\( 0.5|x - 4| = 10 \)
Then, multiply both sides by 2 to solve for \(|x - 4|\):
\( |x - 4|=10\times2 \)
\( |x - 4| = 20 \)

Step3: Solve the absolute value equation

The absolute value equation \(|A| = B\) (where \(B\geq0\)) has two solutions: \(A = B\) or \(A=-B\). So for \(|x - 4| = 20\), we have two cases:

• Case 1: \(x - 4=20\)

Add 4 to both sides: \(x=20 + 4=24\)

• Case 2: \(x - 4=-20\)

Add 4 to both sides: \(x=-20 + 4=-16\)

Answer:

\( x = -16, x = 24 \) (the second option: \( x = -16, x = 24 \))