If the inequality $ (x+2)^{\frac{1}{2}} > x $ is satisfied. what is the range of x ?
My approach – I squared both the sides and proceeded on to solve the quadratic obtained in order to solve the inequality. However, this method only gives -1 and 2 as the roots of the quadratic. But obviously this is not the correct answer as this inequality ( with equality ) even if x is put equal to -2.
What am I doing wrong , why am I obtaining the wrong interval ?
What is the correct method to solve inequalities of this kind ?
Best Answer
HINT
You can't always square inequalities.
You should make sure that both sides are positive. And you also have to make sure the radicand is positive.
However, in this case, the inequality is not to hard to prove that it is true for $-2 \le x<0$, so it does not really matter. Now you merely have to check $x \ge 0$.
Now, you can square both sides and get $$x+2 \ge x^2 \Leftrightarrow (x+1)(x-2) \le 0$$
When faced with problems such as $$f(x) \ge g(x)$$Where both sides involve square roots, you should split it into two cases (Well, maybe not just two.) in order to avoid the above mentioned problem.