The discussion revolves around the process of solving the inequality ##\sqrt{x} < 2x##, specifically focusing on the implications of squaring both sides of the inequality. Participants explore the potential for extraneous solutions that may arise from this operation and the conditions under which it is valid to square both sides.
Participants express varying degrees of caution regarding the squaring operation, with some agreeing on the need to check for extraneous solutions while others emphasize the conditions under which squaring is valid. The discussion remains unresolved regarding a definitive approach to handling such inequalities.
Participants highlight the importance of considering the non-negativity of the expressions involved and the potential for extraneous solutions arising from the squaring process. There is an acknowledgment that the operation of squaring is not reversible, which adds complexity to the solution process.
It's not so much the square root -- it's the simple fact of squaring both sides of an inequality.Mr Davis 97 said:So I am trying to solve a simple rational inequality: ##\sqrt{x} < 2x##. Now, why can't I just square the inequality and go on my way solving what results? What precisely is the reason that I need to be careful when squaring the square root?
Squaring both sides of an equation or inequality is not a one-to-one operation, so isn't reversible. Once you have squared something, you can no longer determine whether the thing that was squared was positive or negative.Mr Davis 97 said:Well, the signs are not an issue in this scenario, but if I don't pay attention to the fact that I am squaring, I get an extraneous result. For example, if we go from ##\sqrt{x} < 2x \implies x < 4x^2 \implies x(4x - 1) > 0##, which then gives us ##x > \frac{1}{4}## or ##x < 0##. However, ##x < 0## is obviously extraneous because of the square root in the beginning of the problem, which implies that x must be positive. Thus, what is it about squaring that causes the extraneous solution set to occur?
So would a hard and fast rule be to just proceed as normal (i.e. square both sides and solve for x), and then after all is said done, check back to see whether you have an extraneous answers?Mark44 said:Squaring both sides of an equation or inequality is not a one-to-one operation, so isn't reversible. Once you have squared something, you can no longer determine whether the thing that was squared was positive or negative.
Leaving aside inequalities for the moment, consider ##\sqrt{x} = -2x##. Squaring both sides gives ##x = 4x^2## or x(4x - 1) = 0, with solutions x = 1/4 or x = 0. x = 1/4 is not a solution of the original equation. This extraneous solution arose because of the squaring operation.
Yes, this is something you should do. This would also include raising both sides to the fourth or other even power.Mr Davis 97 said:So would a hard and fast rule be to just proceed as normal (i.e. square both sides and solve for x), and then after all is said done, check back to see whether you have an extraneous answers?