$xy=10$ vs $xy\geq 10$

Question

What I understand:

$$xy=10\tag{1}$$

Let $x,y$ be two variables. Now, if we increase the value of $x$, then the value of $y$ must decrease, to preserve the validity of $(1)$, and vice versa.

What I don't understand:

$$xy\geq 10$$

Let $x,y$ be two variables. If we increase $x$, I don't think $y$ needs to be decreased, but I'm not sure. Am I right?

Context:

I was studying Heisenberg's Uncertainty Principle, and it is stated below:

$$\Delta x.\Delta p\geq\frac{h}{4\pi}\ [\text{a non-zero positive constant}]$$

$\Delta x$ is the uncertainty in position and $\Delta p$ is the uncertainty in momentum/speed (since mass is constant). My book said that the smaller the $\Delta x$, the larger is the $\Delta p$, and vice versa. I would've had no problem with accepting the previous sentence if Heisenberg's Uncertainty principle was $\Delta x.\Delta p=\frac{h}{4\pi}$, instead of what it is.

Answer 1

Assuming both $x$ and $y$ are nonnegative, and consider the inequality $xy\ge 10$. Then

• if you increase $x$, and $y$ remains the same, then the inequality is still true;

• the smaller $x$ is (than its current value), the larger must $y$ be (than its current value).

No contradiction.

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For example, for $x=5,y=2$, the inequality is true.

• If $x=11$ and $y=2$, the inequality is still true.

• If $x=1$, which is smaller than $5$, $y$ must be at least $10$, which is larger than $5$.