What happens with a sign? I have something simple to ask, but it often confuses me.
What happens with number-sign when it crosses over the equal sign or less-than and greater-than sign. I have an example.
$$1-3x\geq0$$
$$-3x\geq-1$$
$$x\geq-1/?$$ Should there be now $-3$ or just $3$?
And should maybe a sign $>$ change into $<$?
 A: Consider this
You know, $2 \lt 3$ right?
But you also know $-2>-3$ by following the number line, having 0 at the centre, as -2 is closer to 0 than -3. 
Let's divide the number line into positive space $\forall x\gt0$ and negative space $\forall x \lt 0$. Just remember, when you move from one space to another the equality sign will alter, become opposite, for the equality to hold true.
In your case, 
$1-3x \ge0$
Subtract 1 from both sides
$-3x \ge-1$
Now, when you will cancel out the minus sign you are changing/moving from one space to another. So, equality will alter.
$3x \le 1$
$\implies x \le \frac{1}{3}$
A: If you multiply or divide both sides of an inequality by a negative number then you have to flip the sign around, e.g. $$-3x\ge-1\implies x\le-1/(-3) \implies x\le 1/3$$.
If you are adding or subtracting however, you do not need to flip the inequality sign, and this can often make you less uncertain that you are correct by only adding/subtracting rather than multiplying/dividing. For your example,
$$-3x\geq-1\implies 0\ge3x-1\implies1\ge 3x\implies1/3\ge x$$
A: Experiment with different values around $\pm1/3$ and conclude:
$$\begin{align}
&x=-2/3&\to-3x&=2&\color{green}\ge-1\\
&x=-1/3&\to-3x&=1&\color{green}\ge-1\\
&x=-1/6&\to-3x&=1/2&\color{green}\ge-1\\
&x=1/6&\to-3x&=-1/2&\color{green}\ge-1\\
&x=1/3&\to-3x&=-1&\color{green}\ge-1\\
&x=2/3&\to-3x&=-2&\color{red}\ge-1\\
\end{align}$$
