Solve the Rational Equation: 5/(8-x) = 3/(2x)

To solve this problem, we'll follow these steps:

• Step 1: Identify that the given equation is $\frac{5}{8-x} = \frac{3}{2x}$.
• Step 2: Cross-multiply to eliminate the fractions.
• Step 3: Solve the resulting linear equation.
• Step 4: Check for any restrictions on $x$.

Now, let's work through each step:

Step 1: We have the equation:

$\frac{5}{8-x} = \frac{3}{2x}$

Step 2: Cross-multiply to get:

$5 \cdot 2x = 3 \cdot (8-x)$

This simplifies to:

$10x = 24 - 3x$

Step 3: Solve for $x$ by isolating it on one side of the equation. Add $3x$ to both sides:

$10x + 3x = 24$

This simplifies to:

$13x = 24$

Now, divide both sides by 13:

$x = \frac{24}{13}$

Step 4: Verify that this value does not make any of the original denominators zero. For $x = \frac{24}{13}$, the terms $8-x$ and $2x$ are well-defined, and neither is zero:

$8 - \frac{24}{13} = \frac{80}{13} - \frac{24}{13} = \frac{56}{13} \neq 0$

$2 \times \frac{24}{13} = \frac{48}{13} \neq 0$

No issues arise from substituting back, so our solution is valid.

Therefore, the solution to the problem is $x = \frac{24}{13}$, which corresponds to choice 3.