Problem Statement
Solve for $x$: $\frac{1}{x+4} - \frac{1}{x-7} = \frac{11}{30}, \quad x \neq -4, 7$
Verified Solution & Marking Scheme
Combine fractions on LHS
$\frac{(x - 7) - (x + 4)}{(x + 4)(x - 7)} = \frac{11}{30}$
$\frac{x - 7 - x - 4}{x^2 - 7x + 4x - 28} = \frac{11}{30}$
$\frac{-11}{x^2 - 3x - 28} = \frac{11}{30}$
Divide by 11 and cross multiply
$\frac{-1}{x^2 - 3x - 28} = \frac{1}{30}$
$-(30) = x^2 - 3x - 28$
$x^2 - 3x - 28 + 30 = 0 \implies x^2 - 3x + 2 = 0$
Factorize the quadratic equation
$(x - 1)(x - 2) = 0 \implies x = 1 \quad \text{or} \quad x = 2$
Both solutions are valid as $x \neq -4, 7$.