We are solving the equation:

$|4x + 1| + 1 = 3x$

To handle the absolute value, we need to consider two cases: one where $4x + 1 \geq 0$ and another where $4x + 1 < 0$.

Case 1: $4x + 1 \geq 0$

In this case, the absolute value can be removed as $|4x + 1| = 4x + 1$. So, the equation becomes:

$4x + 1 + 1 = 3x$

Simplify:

$4x + 2 = 3x$

Subtract $3x$ from both sides:

$x + 2 = 0$

So, $x = -2$.

We now check if this solution satisfies the condition $4x + 1 \geq 0$:

$4(-2) + 1 = -8 + 1 = -7 \quad (\text{not true, as } -7 < 0)$

Thus, $x = -2$ is not a valid solution.

Case 2: $4x + 1 < 0$

In this case, the absolute value expression becomes $|4x + 1| = -(4x + 1)$, so the equation becomes:

$-(4x + 1) + 1 = 3x$

Simplify:

$-4x - 1 + 1 = 3x$

This simplifies further to:

$-4x = 3x$

Add $4x$ to both sides:

$0 = 7x$

So, $x = 0$.

Now, check if this solution satisfies $4x + 1 < 0$:

$4(0) + 1 = 1 \quad (\text{not true, as } 1 \geq 0)$

Thus, $x = 0$ is not a valid solution.

Conclusion:

There are no valid solutions for this equation.

Do you want details on the cases or have any further questions?

Here are 5 related questions for practice:

1. Solve $|2x - 3| = 5$.
2. Solve $|5x + 4| - 2 = 3x$.
3. Solve $|x + 2| = x + 5$.
4. Solve $|3x + 1| + 2 = 2x$.
5. Solve $|4x - 2| = 6$.

Tip: When solving absolute value equations, always remember to break them into two cases based on whether the expression inside the absolute value is positive or negative.