Solve the following...I keep getting strange answers

$\displaystyle x + \frac{7-2x}{x-2}<= \frac{5x+7}{x-2}$

Solve and express solution as a graph and in interval or set notation.

The solutions are x=0 or x=9. (That's given.)

My work:

he factors I find are x and x-9. Test the factors in the intervals...if their product is - then that interval works. (-inf,0) doesn't work. (0,9) works. (9,inf) doesn't work.

The problem is the interval (-inf,0) works in the original equation.

I'm also confused in general with checking for exceptions, where the function is undefined. Is it done before or after simplifying?

Re: Solve the following...I keep getting strange answers

You obtain

$\displaystyle \frac{x(x-9)}{x-2}\leq 0$

At this point, we're going to make a sign table:

x...| ....... 0 .........2 .........9 ...........

f(x)|........0...........|..........0..........

Take a test point for example $\displaystyle x=1 \Rightarrow f(1)>0$ thus:

x...| ....... 0 .........2 .........9 ...........

f(x)|....-...0....+.....|...-.....0.....+......

Therefore the solution of this inequality is:

$\displaystyle ]-\infty,0]\cup ]2,9]$