Question

solve ( 4|6 - 4x| - 4 = 20 )

Explanation:

Step 1: Isolate the absolute value term

Start with the equation \(4|6 - 4x| - 4 = 20\). First, add 4 to both sides to isolate the absolute value part.
\(4|6 - 4x| - 4 + 4 = 20 + 4\)
\(4|6 - 4x| = 24\)

Step 2: Divide by 4

Divide both sides of the equation \(4|6 - 4x| = 24\) by 4 to simplify.
\(\frac{4|6 - 4x|}{4}=\frac{24}{4}\)
\(|6 - 4x| = 6\)

Step 3: Set up two cases

The absolute value equation \(|A| = B\) (where \(B\geq0\)) is equivalent to \(A = B\) or \(A=-B\). So we have two cases:

• Case 1: \(6 - 4x = 6\)

Subtract 6 from both sides: \(6 - 4x - 6=6 - 6\)
\(-4x = 0\)
Divide by - 4: \(x=\frac{0}{-4}=0\)

• Case 2: \(6 - 4x=-6\)

Subtract 6 from both sides: \(6 - 4x - 6=-6 - 6\)
\(-4x=-12\)
Divide by - 4: \(x = \frac{-12}{-4}=3\)

Answer:

The solutions are \(x = 0\) and \(x = 3\)