Condition in an inequality I have an inequality that is reduced to :
$h(1 - 2v) \geq \frac{1-2v}{2}$
I need to find that :

*

*if $v < 1/2$, then $h \geq 1/2 $

*if $v > 1/2$, then $h \leq 1/2 $
But I am only able to find that  :
$h(1 - 2v) \geq \frac{1-2v}{2}$
$h \geq \frac{1-2v}{(1 - 2v) 2}$
$h \geq \frac{1}{2}$
It may be very simple for someone who is used with mathematics, but I don't see how to solve it. Could you please help me to find these conditions ?
 A: As noticed in the comments, there is a issue with your first step.
We should proceed as follows
$$h (1 - 2v) \geq \frac{1-2v}{2}$$
$$h (1 - 2v) - \frac{1-2v}{2} \ge 0$$
$$(1 - 2v) \left(h-\frac12\right) \ge 0$$
that is

*

*$1 - 2v\ge 0 \iff h-\frac12\ge 0$


*$1 - 2v \le 0\iff h-\frac12\le 0$
A: You begin with an inequality
\begin{align}
h\cdot(1-2v) \geq \frac{1-2v}{2}.
\end{align}
Now the change in sign that you are trying to measure comes from when you divide across the inequality by (1-2v). Notice that if $v > \frac{1}{2}$, then $ 1 - 2v < 0$, i.e. is negative: so denoting by $|1-2v|$ the positive part of $1-2v$
\begin{align}
h\cdot(1-2v) \geq \frac{1-2v}{2} \iff -h\cdot |1-2v| \geq -\frac{|1-2v|}{2}.
\end{align}
Rearranging the inequality on the right hand side then gives us
\begin{align}
h\cdot(1-2v) \geq \frac{1-2v}{2} \iff h\cdot |1-2v| \leq \frac{|1-2v|}{2},
\end{align}
after which we can divide across by the positive $|1-2v|$ to obtain that $h \leq \frac{1}{2}$.
Now if instead $v < \frac{1}{2}$, we find that $1 - 2v > 0$ and so we can perform the division straight away to obtain that $h \geq \frac{1}{2}$.
