How do you solve (3x)/(x+1)+6/(2x)=7/x?

Oct 1, 2016

$x = - \frac{2}{3}$ or $x = 2$

Explanation:

$\frac{3 x}{x + 1} + \frac{6}{2 x} = \frac{7}{x}$ can be rewritten as

$\frac{3 x}{x + 1} = \frac{7}{x} - \frac{6}{2 x}$

or $\frac{3 x}{x + 1} = \frac{7}{x} - \frac{3}{x} = \frac{4}{x}$

Hence, $\frac{3 x}{\left(x + 1\right)} \times x \left(x + 1\right) = \frac{4}{x} \times x \left(x + 1\right)$

or $\frac{3 x}{\cancel{x + 1}} \times x \cancel{\left(x + 1\right)} = \frac{4}{\cancel{x}} \times \cancel{x} \left(x + 1\right)$

or $3 {x}^{2} = 4 x + 4$

or $3 {x}^{2} - 4 x - 4 = 0$

or $3 {x}^{2} - 6 x + 2 x - 4 = 0$

or $3 x \left(x - 2\right) + 2 \left(x - 2\right) = 0$

or $\left(3 x + 2\right) \left(x - 2\right) = 0$

Hence either $3 x + 2 = 0$ i.e. $x = - \frac{2}{3}$

or $x - 2 = 0$ i.e. $x = 2$