We are given the equation:

$\frac{3}{4 - x} = 6$

To solve for $x$, follow these steps:

Step 1: Eliminate the fraction

Multiply both sides of the equation by $4 - x$ to get rid of the denominator:

$3 = 6(4 - x)$

Step 2: Distribute the 6 on the right-hand side

$3 = 24 - 6x$

Step 3: Isolate the $x$-term

Subtract 24 from both sides:

$3 - 24 = -6x$

$-21 = -6x$

Step 4: Solve for $x$

Divide both sides by -6:

$x = \frac{-21}{-6} = \frac{7}{2}$

Thus, the solution is:

$x = \frac{7}{2} \quad \text{or} \quad x = 3.5$

Would you like further clarification or details on any step?

Related questions:

1. How would the solution change if the equation were $\frac{3}{4 - x} = -6$?
2. What happens if the fraction is written as $\frac{3}{x - 4} = 6$?
3. How do we solve $\frac{5}{4 - x} = 7$?
4. What is the solution to $\frac{4}{x + 3} = 2$?
5. How would you solve $\frac{3}{x - 2} = 5x$?

Tip:

Always check if your solution causes any undefined expressions (like division by zero) in the original equation.