The equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ has how many solutions? The equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ has how many solutions?
What would be the correct approach to this problem?Squaring seems to make it even more complicated!
P.S:Sorry,at first I wanted to avoid invalid roots.But thanks for your help.Its a good idea to check at the end.
 A: Like someone said, just do it. The original equation is equivalent to:
$$\frac{2}{\sqrt{x-1}+\sqrt{x+1}}=\sqrt{4x-1} $$
where every term makes sense iff $x\geq 1$. With such assumption, however, the LHS is $\leq\sqrt{2}$ while the RHS is $\geq\sqrt{3}$. So, no solutions.
A: First, we should have $x \ge 1$. Now, the equation can be re-written as:
$$2 = \sqrt{4x - 1} (\sqrt{x + 1} + \sqrt{x - 1})$$
The RHS is $\ge \sqrt3 \times \sqrt2 > 2$. So, the equation has no solutions.
A: You don't really need to calculate with square roots at all: you need only know that $\sqrt x<\sqrt y$ whenever $1\le x<y$. The expression as a whole makes sense if and only if $x\ge 1$, in which case
$$\sqrt{x+1}-\sqrt{x-1}\le\sqrt{x+1}<\sqrt{4x-1}\;,$$
where the last inequality follows from the fact that $1\le x+1<4x-1$. Thus, there is no solution. (In fact $x+1<4x-1$ for $x>\frac{2}3$.)
A: Hint: first note the domain of the equation is the interval $[1,\infty)$. Define $$f(x):=\sqrt{x+1}-\sqrt{x-1}-\sqrt{4x-1}$$
Now note that $f(1)<0$ and $f$ is decreasing.
A: Squaring helps.  You cut three square roots to one, then isolate it and square again. $$\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}\\x+1-2\sqrt{x^2-1}+x-1=4x-1\\
-2\sqrt{x^2-1}=2x-1\\4x^2-4=4x^2-4x+1\\x=\frac 54$$  As we have squared, we have to check the solution we have found, which fails as $\sqrt{\frac 54+1}-\sqrt{\frac 54-1}=\frac 32-\frac 12=1 \neq \sqrt{4 \cdot \frac 54-1}=2$ .  There is no solution.  This is not surprising.  The two terms on the left are not very different, so their difference will be too small to equal the term on the right.
A: No solution.
Squaring actually does not make it much worse! If you square both sides you get.
$(x + 1) + (x - 1) - 2(\sqrt{x^2 - 1}) = 4x - 1$
Rearrange this to get. 
$-2\sqrt{x^2 - 1} = 2x - 1$
Square both sides now. You'll be left with a linear equation in x.
When you solve it, you'll get 5/4, which won't satisfy the original equation. So nothing more to be done.
