We are given the equation:

$7x + x = 4x + (x + 1)$

Step 1: Simplify both sides

On the left side:
$7x + x = 8x$
On the right side:
$4x + (x + 1) = 4x + x + 1 = 5x + 1$

Now, the equation simplifies to:
$8x = 5x + 1$

Step 2: Isolate $x$

Subtract $5x$ from both sides:
$8x - 5x = 5x - 5x + 1$
$3x = 1$

Step 3: Solve for $x$

Divide both sides by 3:
$x = \frac{1}{3}$

Final Answer:

$\boxed{\frac{1}{3}}$

Would you like a more detailed explanation or another example?

Here are some related questions:

1. What happens if the equation has no solution?
2. How do you solve equations with fractions?
3. Can you solve an equation with variables on both sides and parentheses?
4. How do you check if your solution is correct?
5. What is an identity equation, and how does it differ from a standard equation?

Tip: Always check your solution by substituting it back into the original equation! 🚀