Question

The absolute value equation has 3 | a − 4 | = 12 has two solutions. The solutions can be represented as the solutions to which of the following pairs?

Answer 1

To solve the equation \( 3 | a - 4 | = 12 \), we first isolate the absolute value term:

\[
| a - 4 | = \frac{12}{3} = 4.
\]

Now, we can set up the two cases for the equation inside the absolute value:

1. \( a - 4 = 4 \)
2. \( a - 4 = -4 \)

Now we solve each case:

**Case 1:**
\[
a - 4 = 4 \\
a = 4 + 4 = 8.
\]

**Case 2:**
\[
a - 4 = -4 \\
a = -4 + 4 = 0.
\]

Thus, the two solutions are \( a = 8 \) and \( a = 0 \). Therefore, the solutions can be represented as the pair \( (0, 8) \).