Question

$6 \div x + 1 \: + 5 \div 2x + 1 \: = 3$

Answer 1

Answer -

$\frac{6}{x + 1} + \frac{5}{2x + 1} = 3 \\ \\ \frac{6(2x + 1) + 5(x + 1)}{(x + 1)(2x + 1)} = 3 \\ \\ \frac{12x + 6 + 5x + 5}{2x {}^{2} + x + 2x + 1 } = 3 \\ \\ \frac{17x + 11}{2x {}^{2} + 3x + 1} = 3 \\ \\ \bf on \: cross - multiplying : \\ \\ 3(2x {}^{2} + 3x + 1) = 17x + 11 \\ \\ 6x {}^{2} + 9x + 3 = 17x + 11 \\ \\ 6x {}^{2} + 9x - 17x + 3 - 11 = 0 \\ \\ 6x {}^{2} - 8x - 8 = 0$

$\sf Now, \: we \: got \: quadratic \: equation: \\ \\ 6x {}^{2} - 8x - 8 = 0 \\ \\ 2(3x {}^{2} - 4x - 4) = 0 \\ \\ 3x {}^{2} - 4x - 4 = 0 \\ \\ 3x {}^{2} - 6x + 2x - 4 = 0 \\ \\ 3x(x - 2) + 2(x - 2) = 0 \\ \\ (x - 2)(3x + 2) = 0 \\ \\ x - 2 = 0 \: | 3x + 2 = 0 \\ x = 2 \: \: \: \: \: \: \: \: \: | x = \frac{ - 2}{3}$

$\bf Therefore.. \\ \\ \boxed{\bf x = 2 \: or \: x = \frac{ - 2}{3}}$