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**Reckoner** $\displaystyle \frac{|x^2+5x+6|}{x^2-9}$

$\displaystyle = \frac{|(x + 2)(x + 3)|}{(x + 3)(x - 3)}$

$\displaystyle = \frac{|x + 2||x + 3|}{(x + 3)(x - 3)}$

Now, the absolute value will depend on the values that $\displaystyle x$ is taking. As $\displaystyle x\to a-$, is $\displaystyle x + 3$ positive or negative? What about $\displaystyle x + 2$ and $\displaystyle x - 3$? How about as $\displaystyle x\to a+$? This will all, of course, depend on the value of $\displaystyle a$.

When the expression is positive, you can remove the absolute bars, and when it is negative you can remove the bars as long as you negate the expression.