How to solve the following equation: $\frac{3}{x} - \frac{x-3}{2x+10} + \frac{8}{3x+15} = \frac{4}{3}$?

Question

I'm facing difficulties in solving the following equation - would someone mind giving me a hint?

$$\frac{3}{x} - \frac{x-3}{2x+10} + \frac{8}{3x+15} = \frac{4}{3}$$

Thanks in advance!

Answer 1

Hint: $2x+10$ and $3x+15$ have what in common?

Further hint: just like fractions with numbers, fractions with expressions involving variables can be recombined using algebra. For instance,

$$\frac ab - \frac c{de}+\frac f{dg}=\frac hi$$

can be rewritten by multiplying through by each of the denominators, minus the duplicate factors -- in this case, $de,dg$ share a factor $d$, so we would multiply by $bdegi$:

$$adgi - cbgi+fbei=hbdeg$$

However, care must be taken to check that $b,de,dg,i\neq0$.

Answer 2

Hint: $$\frac{3}{x} - \frac{x-3}{2x+10} + \frac{8}{3x+15} = \frac{4}{3}$$ $$\frac{18}{x} - \frac{3x-25}{x+5} = 8$$ $$18(x+5)-(3x-25)x=8x(x+5)$$

But remember, $x\neq 0,-5$.