The solution of inequality (x+1)(x^{2+1)1/2>x2-1 is}

(a) {-1} U [2,infinite)

(b) [-1,infinite)

(c){-4}u[2,3]

(d)(-5,infinite)

Hi,

I am assuming your question is (x+1) sqrt(x2+1) > x2-1.

This can be re-written as (x+1) sqrt(x2+1) > (x+1)(x-1)

Case 1 : (x+1)>0, or x > -1

In this case, we can cancel (x+1) in both sides.

We get sqrt(x2+1)>(x-1)

This is valid for all x>(-1). So Case 1 gives a solution x> -1

Case 2: (x+1)<0 or x < -1

We can still cancel (x+1) from both sides, but since (x+1) is negative, we have to change the > sign to <.

SO now we have sqrt(x2+1) < (x-1).

This is not possible because x-1< -2 (because x< -1 in this case).

And sqrt() will always be positive, so it can never be less than -1.

Hence Case 2 gives no results.

Therefore the answer is x > -1 or x = (-1,inf). This is not mentioned in any of the options so all options are INCORRECT.

You can verify this by putting x=-1 and x=-3. Putting x=-1, we get LHS=0 and RHS=0, SoLHS>RHS is not true. So x=-1 is not a solution. So options A and B are wrong. Similarly, we find that x=-3 soes NOT satisfy the equation, so options C and D are also wrong. 

Correct range is (-1,inf)