Inequality Problem Related to a Spivak Problem (Ch 1 16(b))

Question

This question came up when I was looking at a Spivak problem, but I just want to check the step below. Given $$4{x}^{2} + 8xy + 4{y}^{2} \ge 0 $$

and
$$4{x}^{2} + 6xy + 4{y}^{2} \le 0$$

"Subtract the second equation from the first."

Is this the correct approach for this particular step?

$$4{x}^{2} + 6xy + 4{y}^{2} \le 0 \implies -(4{x}^{2} + 6xy + 4{y}^{2}) \ge 0$$

Now $$4{x}^{2} + 8xy + 4{y}^{2} \ge 0 \land -(4{x}^{2} + 6xy + 4{y}^{2}) \ge 0 \implies 2xy \ge 0 $$

Thanks in advance.

Answer 1

It is "correct", but you should not use "=" for $\implies$. This is typed with \implies. It is correct because you applied the rule $$ a\ge 0 , b\ge 0 \implies a+b\ge 0 .$$

For reference, Spivak's solution is:

(b) The first equation implies that $$ 4 x^{2}+8 x y+4 y^{2} \geq 0 $$ Suppose that we also had $$ 4 x^{2}+6 x y+4 y^{2} \leq 0 $$ Subtracting the second from the first would give $2 x y \geq 0$. If neither $x$ nor $y$ is 0 , this means that we must have $2 x y>0$; but this implies that $4 x^{2}+6 x y+y^{2}>0$, a contradiction.

Moreover, it is clear that if one of $x$ and $y$ is 0 , but not the other, then we also have $4 x^{2}+6 x y+4 y^{2}>0$