The given equation is:

\[
|3x + 1| + 5 = x
\]

To solve this equation, we need to consider two cases for the absolute value:

### Case 1: \(3x + 1 \geq 0\)
In this case, \( |3x + 1| = 3x + 1 \), so the equation becomes:
\[
3x + 1 + 5 = x
\]
Simplifying:
\[
3x + 6 = x
\]
Subtract \(x\) from both sides:
\[
2x + 6 = 0
\]
Subtract 6 from both sides:
\[
2x = -6
\]
Divide by 2:
\[
x = -3
\]

We need to check if this solution satisfies \(3x + 1 \geq 0\):
\[
3(-3) + 1 = -9 + 1 = -8
\]
Since \(-8 < 0\), \(x = -3\) does **not** satisfy the condition, so this solution is **not valid**.

### Case 2: \(3x + 1 < 0\)
In this case, \( |3x + 1| = -(3x + 1) \), so the equation becomes:
\[
-(3x + 1) + 5 = x
\]
Simplifying:
\[
-3x - 1 + 5 = x
\]
\[
-3x + 4 = x
\]
Add \(3x\) to both sides:
\[
4 = 4x
\]
Divide by 4:
\[
x = 1
\]

We need to check if this solution satisfies \(3x + 1 < 0\):
\[
3(1) + 1 = 3 + 1 = 4
\]
Since \(4 > 0\), \(x = 1\) does **not** satisfy the condition, so this solution is **also not valid**.

### Conclusion:
There is no solution to the equation \(|3x + 1| + 5 = x\).

Would you like more details or have any questions? Here are some related questions:

1. How do we determine when an equation has no solution?
2. Can absolute value equations have multiple solutions?
3. What happens if the absolute value expression is more complicated?
4. How do inequalities factor into solving absolute value equations?
5. How can this type of equation be solved graphically?

**Tip**: When solving absolute value equations, always check the solutions with the conditions imposed by the absolute value.

Learn how to solve the absolute value equation |3x + 1| + 5 = x with a detailed step-by-step explanation. This solution covers key algebraic concepts like absolute value and case analysis, suitable for high school students in Grades 10-12.

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