To solve the equation [tex]\( -3|x - 3| = -6 \)[/tex], let's go through the steps in a detailed manner:
1. First, isolate the absolute value term. We can do this by dividing both sides of the equation by [tex]\(-3\)[/tex]:
[tex]\[
-3|x - 3| = -6
\][/tex]
[tex]\[
|x - 3| = \frac{-6}{-3}
\][/tex]
[tex]\[
|x - 3| = 2
\][/tex]
2. The equation [tex]\( |x - 3| = 2 \)[/tex] means that the expression inside the absolute value can be either 2 or -2. Therefore, we set up two separate equations to solve for [tex]\( x \)[/tex]:
[tex]\[
x - 3 = 2 \quad \text{or} \quad x - 3 = -2
\][/tex]
3. Solving each equation separately:
- For [tex]\( x - 3 = 2 \)[/tex]:
[tex]\[
x - 3 = 2
\][/tex]
[tex]\[
x = 2 + 3
\][/tex]
[tex]\[
x = 5
\][/tex]
- For [tex]\( x - 3 = -2 \)[/tex]:
[tex]\[
x - 3 = -2
\][/tex]
[tex]\[
x = -2 + 3
\][/tex]
[tex]\[
x = 1
\][/tex]
4. Therefore, the solutions to the equation [tex]\( -3|x - 3| = -6 \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = 5 \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{x = 1, x = 5}
\][/tex]
