Solve for $x$: $\dfrac{1}{x+4} - \dfrac{1}{x-7} = \dfrac{11}{30}$, where $x
eq -4, 7$.
Step-by-Step Solution
Key Concept: Take common denominator on LHS, simplify, cancel factors, and solve resulting quadratic.
Stepwise Solution:
$\dfrac{(x - 7) - (x + 4)}{(x + 4)(x - 7)} = \dfrac{11}{30} \Rightarrow \dfrac{-11}{x^2 - 3x - 28} = \dfrac{11}{30}$. [1.0 Mark]
Divide both sides by $11$: $\dfrac{-1}{x^2 - 3x - 28} = \dfrac{1}{30} \Rightarrow x^2 - 3x - 28 = -30$. [1.0 Mark]
$x^2 - 3x + 2 = 0 \Rightarrow (x - 1)(x - 2) = 0 \Rightarrow x = 1$ or $x = 2$. Solution: $x = 1, 2$. [1.0 Mark]
Marking Scheme:
• LHS fraction subtraction and simplification: 1.0 Mark
• Forming quadratic equation $x^2 - 3x + 2 = 0$: 1.0 Mark
• Factorising and finding roots $x = 1, x = 2$: 1.0 Mark
Correct Answer: