Question

Let P = \(\frac{2x}{x+3}-\frac{x+1}{x-3}+\frac{3-11x}{9-x^2}\). Find the smallest integer x such that P is also an integer.

Answer 1

\(x\ne\pm3\)

\(P=\frac{2x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}-\frac{\left(x+1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}+\frac{11x-3}{\left(x-3\right)\left(x+3\right)}\)

\(=\frac{x^2+x-6}{\left(x-3\right)\left(x+3\right)}=\frac{\left(x+3\right)\left(x-2\right)}{\left(x+3\right)\left(x-3\right)}\)

\(=\frac{x-2}{x-3}=1+\frac{1}{x-3}\)

P is an integer if and only if 1 is divisible by \(x-3\)

Therefore \(x-3=\left\{-1;1\right\}\Rightarrow x=\left\{2;4\right\}\)

\(\Rightarrow x_{min}=2\)