Let's solve the equation step by step:

$\frac{4x}{7} - \frac{1}{2} = -\frac{3}{4} - \frac{2x}{5}$

Step 1: Eliminate fractions by finding the least common denominator (LCD).

The denominators are 7, 2, 4, and 5. The least common denominator for these is 140.

Multiply each term by 140 to eliminate the fractions:

$140 \cdot \left( \frac{4x}{7} \right) - 140 \cdot \left( \frac{1}{2} \right) = 140 \cdot \left( -\frac{3}{4} \right) - 140 \cdot \left( \frac{2x}{5} \right)$

Simplifying each term:

$\frac{140}{7} \cdot 4x - \frac{140}{2} \cdot 1 = \frac{140}{4} \cdot (-3) - \frac{140}{5} \cdot 2x$

$20 \cdot 4x - 70 = -35 \cdot 3 - 28 \cdot 2x$

Now calculate:

$80x - 70 = -105 - 56x$

Step 2: Move all terms involving $x$ to one side and constants to the other.

Add $56x$ to both sides:

$80x + 56x = -105 + 70$

Simplifying:

$136x = -35$

Step 3: Solve for $x$.

$x = \frac{-35}{136}$

So the solution is:

$x = -\frac{35}{136}$

Would you like further clarification on any steps, or have any questions?

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Here are 5 related questions you might find interesting:

1. How do you solve equations with more than one variable?
2. How do you handle equations involving fractions or decimals?
3. What is the least common denominator, and why is it useful?
4. How can you check if your solution to an equation is correct?
5. How do you deal with negative fractions in algebraic equations?

Tip: When solving equations with fractions, always start by eliminating the fractions to simplify the equation. It often makes the process quicker and easier.