# How can I determine the behavior of a certain inequality? [closed]

Consider the expression

$$\qquad 2(1-2c)/(c-1)^2$$

where $$c < 0$$.

I would like to know if it is bigger than 1 anywhere in its domain.

2(1-2c)/(c-1)^2

• If you evaluate Reduce[{2 (1 - 2 c) / (c - 1)^2 > 1, c < 0}], you will see that your expression is larger than 1 only in a certain range of negative values of $c$. Sep 9, 2020 at 4:30

## 1 Answer

The function range of your expression (let's say, $$y$$) is $$0 < y < 2$$, which can be shown by

FunctionRange[{2 (1 - 2 c) / (c - 1)^2, c < 0}, c, y]


To get the point where $$y = 1$$,

Solve[2 (1 - 2 c) / (c - 1)^2 == 1 && c < 0, c]


For completeness: as already MarcoB mentioned in his comment, you can get the range where $$y > 1$$ (or $$y < 1$$) by Reduce:

Reduce[2 (1 - 2 c) / (c - 1)^2 > 1 && c < 0]

Reduce[2 (1 - 2 c) / (c - 1)^2 < 1 && c < 0]