To solve the equation \( |3x + 8| + 4 = x \), let's break it down into steps:

### Step 1: Isolate the absolute value
We start by isolating the absolute value expression on one side of the equation.

\[
|3x + 8| = x - 4
\]

### Step 2: Consider the two cases for the absolute value
An absolute value equation has two cases: one where the expression inside the absolute value is positive (or zero), and one where it's negative.

#### Case 1: \( 3x + 8 \geq 0 \)
In this case, the absolute value doesn't affect the expression, so:
\[
3x + 8 = x - 4
\]
Solving for \( x \):
\[
3x - x = -4 - 8
\]
\[
2x = -12
\]
\[
x = -6
\]
We need to check if this solution satisfies \( 3x + 8 \geq 0 \):
\[
3(-6) + 8 = -18 + 8 = -10 \quad (\text{not true, so discard this solution}).
\]

#### Case 2: \( 3x + 8 < 0 \)
In this case, the absolute value negates the expression inside, so:
\[
-(3x + 8) = x - 4
\]
Solving for \( x \):
\[
-3x - 8 = x - 4
\]
\[
-3x - x = -4 + 8
\]
\[
-4x = 4
\]
\[
x = -1
\]
Check if this solution satisfies \( 3x + 8 < 0 \):
\[
3(-1) + 8 = -3 + 8 = 5 \quad (\text{not true, so discard this solution}).
\]

### Conclusion
There are no valid solutions to this equation. Thus, the equation has **no solution**.

Would you like to discuss further details or ask any questions?

Here are some related questions for further practice:

1. Solve \( |2x - 5| = 3x + 7 \).
2. Solve \( |x + 1| + 3 = 5 \).
3. Solve \( 4 + |x - 2| = 10 \).
4. Solve \( |x^2 - 9| = 16 \).
5. Solve \( |2x + 3| = |x - 5| \).

**Tip**: Always check the conditions for validity when solving absolute value equations to avoid extraneous solutions.

Learn how to solve the absolute value equation |3x + 8| + 4 = x with a step-by-step breakdown. This algebraic problem covers the key concepts of isolating the absolute value and considering different cases. Suitable for students in Grades 9-10.

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