Solve the below inequality :-
$\displaystyle
{(x^2 - 2x - 3)}^{1/2} < 1
$
Any help would be appreciated.
Thanks
The square root of any number larger than 1 is itself larger than 1.
That means that to have square root less than 1, the number itself must be less than 1.
So we have to solve $\displaystyle x^2- 2x- 3< 1$ which is the same as $\displaystyle x^2- 2x< 4$.
Adding 1 to both sides, to "complete the square", $\displaystyle x^2- 2x+ 1< 5$ which is the same $\displaystyle (x-1)^2< 5$. Now, $\displaystyle (x-1)^2= 5$ when $\displaystyle x- 1= \pm\sqrt{5}$ or $\displaystyle x= 1\pm\sqrt{5}$. Since x= 0, which is between those values, satisfies the inequality, all numbers between them and only numbers between them satisfy the inequality. (For continuous functions, inequalities can only reverse at points where they are [b]equalties.)