# How to read the symbol $\gtreqqless$?

The book I am currently reading, is written in a local language, uses the symbol $$\gtreqqless$$

Noted usage in the book (translated):

Let $$f(x) = x^3e^{-3x}$$ for x > 0. Then what is the maximum value of $$f(x)$$ ?

Solution : $$\dotsc$$ $$f'(x) = 3e^{-3x}x^2(1-x)$$... We have to work explicitly with $$1 - x$$ since $$3e^{-3x}$$ is positive for x > 0 $$\therefore$$ We have to see when 1 - x $$\geqslant$$ 0 and when 1 - x $$\leqslant$$ 0. We can see that 1 - x $$\gtreqqless$$ 0 $$\Longleftrightarrow$$ 1 $$\gtreqqless$$ x $$\dotsc$$

So how is the symbol read ,or, What does it mean ?

Edit: I have searched through the book and it does not provide any page on symbols used and no past explanation has been provided.

• It just means that if you place any of the three symbols ($=,>,<$) at both equations, the equivalence holds. It is basically a shortcut for writing $1-x>0\Leftrightarrow 1>x,\ 1-x=0\Leftrightarrow 1=x$ and $1-x<0\Leftrightarrow 1<x$. Jul 3, 2021 at 15:49
• @StefanAlbrecht I see. Feel free to add it as an answer so I can accept it. Jul 3, 2021 at 15:52
• There are two answers already, which essentially convey the same thing, you can accept either of them (your preference), or if @StefanAlbrecht wishes, he can post one. Jul 3, 2021 at 16:00
• It is not too difficult to guess what it means, but how you read it is another story... I would be interested to have the opinion of a native English speaker... Jul 3, 2021 at 16:25
• @J.-E.Pin One possible reading is, "One minus $x$ is greater than, equal to, or less than zero if and only if one is (respectively) greater than, equal to, or less than $x.$" Jul 3, 2021 at 17:12

It is similar to $$\pm$$, so for example, $$(a\pm b)^2=a^2\pm 2ab+b^2$$, if you take a plus you take it everywhere, if you take a minus, you take it everywhere.

Likewise, if you consider $$1-x>0$$, you have $$1>x$$. (Here, we took the topmost symbol!)

If you have $$1-x=0$$, it means $$1=x$$; here we took the second symbol $$=$$. You can see the pattern now.

Hope this helps. Ask anything if not clear :)

• Thank you for the explanation as well! Jul 3, 2021 at 16:05

It's a shorthand notation for $$1-x > 0 \iff 1 > x \quad\text{and}\quad 1-x = 0 \iff 1 = x \quad\text{and}\quad 1-x < 0 \iff 1 < x \, .$$

• Thank you for the answer :) Jul 3, 2021 at 16:05