Question - Solving a Rational Equation

Given equation:

\(\frac{x}{2} = \frac{5x - 24}{x - 4}\)

To solve for \( x \), cross-multiply to eliminate the denominators:

\(x(x - 4) = 2(5x - 24)\)

Expand both sides:

\(x^2 - 4x = 10x - 48\)

Move all terms to one side to set the equation to zero:

\(x^2 - 4x - 10x + 48 = 0\)

Simplify by combining like terms:

\(x^2 - 14x + 48 = 0\)

Factor the quadratic equation:

\((x - 6)(x - 8) = 0\)

Set each factor equal to zero:

\(x - 6 = 0 \quad \text{or} \quad x - 8 = 0\)

Solve for \( x \):

\(x = 6 \quad \text{or} \quad x = 8\)