Domain of a function quick question $$f(x)=\arcsin\left(\frac{x-1}{2x}\right)$$
We need to find the domain of this function.
My try:
$$-1\leq \frac{x-1}{2x} \leq 1 $$
We can split this into
$$-1\leq \frac{x-1}{2x} \quad\text{and} \quad\frac{x-1}{2x} \leq 1$$
My idea is to solve for this two inequalities and then take the intersection of them both.
$$-2x\leq x-1\\
-3x\leq -1\\
x\geq\frac{1}{3}$$
$$x-1 \leq 2x\\
-1 \leq x$$
By doing the intersection I get the wrong answer which is: $x\geq\frac{1}{3}$
What have I done wrong?

Hint:
You forgot the case when x is negative and then, you can't multiply the inequality with $2x$.You need to change the direction of the inequality.
So take by the 2 cases:
1.$x>0$
2.$x<0$
If you still don't get it to work, tell me and I will come up with a complete solution
