Question

Solve the equation 6$(\frac{2x+5}{x+1} )-4(\frac{x+1}{2x+5} )-5=0.$

Answer 1

SOLUTION:-

Let 2x+5/x+1= y

Therefore,

The equation can be written as;

$6y - 4 \times \frac{1}{y } - 5 = 0 \\ \\ 6 {y}^{2} - 5y - 4 = 0 \\ \\ 6 {y}^{2} - 8y + 3 y - 4 = 0 \\ \\ 2y(3y - 4) + 1(3y - 4) = 0 \\ \\ (3y - 4)(2y + 1) = 0 \\ \\ 3y - 4 = 0 \: \: or \: \: 2y+ 1 = 0 \\ \\ 3y = 4 \: \: or \: \: 2y = - 1 \\ \\ y = \frac{4}{3} \: \: or \: \: y = - \frac{1}{2}$

The value of y is 4/3

If x= 4/3

$\frac{2x + 5}{x + 1} = \frac{4}{3} \\ (cross \: multiplication) \\ \\ 6x + 15 = 4x + 4 \\ \\ 6x - 4x = 4 - 15 \\ \\ 2x = - 11 \\ \\ x = \frac{ - 11}{2}$

&

y= -1/2, then

$\frac{2x + 5}{x + 1 } = - \frac{1}{2} \\ \\ 4x + 10 = - x - 1 \\ \\ 4x +x = - 1 - 10 \\ \\ 5x = - 11 \\ \\ x = \frac{ - 11}{5}$

Thus,

Solution set of given equation is(-11/2,-11/5).