TITLE: Squaring both sides of an inequality of opposite signs
QUESTION [16 upvotes]: Can someone explain to why the algebra does not work for this inequality
$\sqrt{1-y}\gt-\frac{1}{2} $.
$1-y\gt\frac{1}{4} $.
$y\lt\frac{3}{4} $.
I know that it should be $y \lt 1$ since the RHS is $\ge0$

REPLY [5 votes]: This is the difference between equivalence and implication.
You can do operations that preserve equivalence, which addition, subtraction, division multiplication and monotonic functions do
$$x+1 = x^2+2x\Leftrightarrow\\0 = x^2+x-1$$
Then all solutions are preserved and no new solutions are added.
Implication on the other hand is only one way. We are sure to not lose solutions, but not sure to not introduce new ones. Squaring is such an operation. In fact for complex numbers squaring automatically doubles the number of solutions for polynomials.