Question

Solve the equation for all values of x.
open vertical bar, 3, x, minus, 1, close vertical bar, minus, 6, equals, 2, x
∣3x−1∣−6=
2x

Answer 1

To solve the equation \( |3x - 1| - 6 = 2x \), we will first isolate the absolute value:

\[
|3x - 1| = 2x + 6
\]

This gives us two cases to consider based on the definition of absolute value.

### Case 1: \( 3x - 1 = 2x + 6 \)

We can solve this equation:

\[
3x - 1 = 2x + 6
\]
Subtract \(2x\) from both sides:

\[
3x - 2x - 1 = 6
\]
Which simplifies to:

\[
x - 1 = 6
\]

Adding 1 to both sides gives us:

\[
x = 7
\]

### Case 2: \( 3x - 1 = -(2x + 6) \)

Now we solve this case:

\[
3x - 1 = -2x - 6
\]
Adding \(2x\) to both sides:

\[
3x + 2x - 1 = -6
\]
This simplifies to:

\[
5x - 1 = -6
\]

Adding 1 to both sides gives us:

\[
5x = -5
\]

Dividing both sides by 5 gives:

\[
x = -1
\]

### Summary of Solutions

The possible solutions from the cases are \(x = 7\) and \(x = -1\).

### Verification of Solutions

1. For \(x = 7\):

\[
|3(7) - 1| - 6 = 2(7)
\]
Calculating the left side:

\[
|21 - 1| - 6 = |20| - 6 = 20 - 6 = 14
\]

Calculating the right side:

\[
2 \cdot 7 = 14
\]

Both sides are equal, so \(x = 7\) is a valid solution.

2. For \(x = -1\):

\[
|3(-1) - 1| - 6 = 2(-1)
\]
Calculating the left side:

\[
|-3 - 1| - 6 = |-4| - 6 = 4 - 6 = -2
\]

Calculating the right side:

\[
2 \cdot (-1) = -2
\]

Both sides are equal, so \(x = -1\) is also a valid solution.

### Final Solutions

The solutions to the equation \( |3x - 1| - 6 = 2x \) are:

\[
\boxed{7 \text{ and } -1}
\]