We start by solving the equation:

$3 - \frac{4x}{5} = 4 + \frac{x}{5}$

Step 1: Eliminate fractions

To eliminate the fractions, multiply through by 5 (the least common denominator):

$5 \cdot \left(3 - \frac{4x}{5}\right) = 5 \cdot \left(4 + \frac{x}{5}\right)$

This simplifies to:

$15 - 4x = 20 + x$

Step 2: Rearrange terms

Move all terms involving $x$ to one side and constants to the other:

$-4x - x = 20 - 15$

Simplify:

$-5x = 5$

Step 3: Solve for $x$

Divide both sides by $-5$:

$x = -1$

Final Answer:

$x = -1$

Let me know if you would like to see further verification or details.

Related Questions:

1. How do you check if the solution satisfies the original equation?
2. What are the properties of equations with fractions?
3. How do you solve equations where variables appear in both numerator and denominator?
4. Can this equation be solved graphically? If so, how?
5. How can you rewrite this equation to model a real-world scenario?

Tip:

When solving equations with fractions, always aim to eliminate the fractions early by multiplying through by the least common denominator.