Question

\\(\frac{7}{x + 1} = \frac{2x - 1}{36}\\)

Explanation:

Step1: Cross-multiply to eliminate fractions

$7 \times 36 = (2x - 1)(x + 1)$

Step2: Calculate left side, expand right side

$252 = 2x^2 + 2x - x - 1$

Step3: Simplify the quadratic equation

$2x^2 + x - 253 = 0$

Step4: Solve quadratic via factoring/quadratic formula

Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a=2$, $b=1$, $c=-253$
$\sqrt{1^2 - 4 \times 2 \times (-253)} = \sqrt{1 + 2024} = \sqrt{2025} = 45$
$x = \frac{-1 \pm 45}{4}$

Step5: Compute valid solution

Discard negative solution (avoids zero denominator):
$x = \frac{-1 + 45}{4} = \frac{44}{4} = 11$

Answer:

$x = 11$