Question

The number of solutions of $\sqrt{x+1}-\sqrt{x-1}=1(x\in R)$ is

• A. 1
• B. 2
• C. 4
• D. infinite

Answer 1

Answer: (A)

For the equation to make sense
we must have $x+1\ge 0$ and $x-1\ge 0\Rightarrow x\ge -1,x\ge 1$
i.e. $x\ge 1$.
We rewrite equation as
$\sqrt{x+1}=1+\sqrt{x-1}$
and square both the sides to obtain
$x+1=1+x-1+2\sqrt{x-1}$
$\Rightarrow \frac{1}{2}=\sqrt{x-1}\Rightarrow \frac{1}{4}=x-1\text{ or }x=\frac{5}{4}$
Also, $x=5/4$ satisfies the given equation.