This is my first question on the site and I'm sure there'll be many more. But for now to the point.
I am really having trouble understanding the proper use of roots and exponents to try to solve inequalities. The thing is that I have noticed that using such operations on many occasions do not produce an equivalent inequation.
My problems started with the following excercise $\sqrt{8x^{2}+22x+15} > 4x+3$
I first considered that $\ 8x^{2}+22x+15\geq 0$, so I know that root expression will be a real number as long as $\ x \leq-3/2$ OR $\ x \geq-5/4$
My attempted solutions:
1) Square both sides to obtain a second-degree polynomial on each side, simplify and solve.
$$\ \big( \sqrt{8x^{2}+22x+15} \big) ^{2} > \big(4x+3\big) ^{2} $$
RESULT: obtained the wrong solution and I'm under the impression that squaring doesn't produce an equivalent inequality.
2) Subtract $\ 4x + 3 $ on both sides and try to eliminate the square roots by squaring both sides.
$$\ \big(\sqrt{8x^{2}+22x+15} - \big(4x+3\big)\big)^{2} > 0 $$ $$\ 24x^2+46x+12-2\big(4x+3\big)\sqrt{8x^{2}+22x+15} > 0 $$
If I attempt this $\ 24x^2+46x+12 > 2\big(4x+3\big)\sqrt{8x^{2}+22x+15} $ and square both sides to eliminate the square root I will get the wrong answer in the end since I know that this does not produce an equivalent inequation. (Tried solving like this before so I kinda figured it was a wrong move.)
I would like an insight as to how to solve these kind of problems and what is the correct way to do them. What mathematical facts or rules am I missing here? How should I treat roots and exponents in general?
Help is very much appreciated!