Domain of $\sqrt{1+\frac1x}$ I solved this as follows:
$1+\frac1x \ge 0$
$\frac1x \ge -1$ (subtract $1$ from both sides)
$1 \ge -x$ (multiply $x$ to both sides, cancel out $x$ from bottom of left side)
$-1 \le x$ (multiply both sides by $-1$, change sign)
$x \ge -1$
Another way to solve is:
$1+\frac1x \ge 0$
$\frac{(x+1)}x \ge 0$
$x+1 \ge 0$
$x \ge -1$
The domain should be $x\ge -1$, not including $0$.  However, I see from WolframAlpha and searching online that the domain is $x\le -1$ or $x>0$.  Why is my solution wrong?

You can't multiply by $x$ unless you take care to note if $x > 0$ or $x < 0$.  Separate the cases and you can get a correct analysis.  You know right off that $x = 0$ is not in the domain; you can exclude it immediately.
