Solving Absolute Value Equation: |-x+6| = |3x-2|

To solve the problem, follow these steps:

• Case 1: Set $-x + 6 = 3x - 2$.
• Simplify and solve for $x$:

$-x + 6 = 3x - 2$
Add $x$ to both sides: $6 = 4x - 2$
Add 2 to both sides: $8 = 4x$
Divide both sides by 4: $x = 2$

• Case 2: Set $-x + 6 = -(3x - 2)$, which is $-x + 6 = -3x + 2$.
• Simplify and solve for $x$:

$-x + 6 = -3x + 2$
Add $3x$ to both sides: $2x + 6 = 2$
Subtract 6 from both sides: $2x = -4$
Divide both sides by 2: $x = -2$

Finally, verify that both solutions satisfy the original absolute value equation:

• When $x = 2$: $|-2 + 6| = |6 - 2| \Rightarrow |4| = |4|$, which holds true.
• When $x = -2$: $|2 + 6| = |-6 - 2| \Rightarrow |8| = |8|$, which holds true.

Thus, both $x = 2$ and $x = -2$ are valid solutions to the equation.

The solutions to the problem are $\boldsymbol{x=-2}$ and $\boldsymbol{x=2}$.

Therefore, the correct answer choice is $\boldsymbol{x=-2}$ , $\boldsymbol{x=2}$.