Inverting price functions and their conditions/intervals If I have demand functions

For $P&lt;15$:

$$
Q(P) = 700-40P
$$

For $P&gt;15$:
$$
Q(P) = 400-20P
$$

If I invert them, I get the price functions

For $Q&lt;100$:
$$
P(Q)=20 - (1/20)Q
$$

For $Q&gt;100$:
$$
P(Q)=17,5 – (1/40)Q
$$

I know how I get the expressions, for instance, $Q = 700 - 40P \Leftrightarrow 40P = 700 - Q \Leftrightarrow P = \frac{700}{40} - \frac{1}{40} Q$, but I don't how the conditions $P &lt; 15$ can change to $Q &lt; 100$.
Notice that if you plug in $P=15$ into either of the $Q(P)$, you get

$$Q(P) = 700-40P = 700 - 40 \cdot 15 = 100$$

$$Q(P) = 400-20P = 400 - 20 \cdot 15 = 100$$

So you can take the exercise further and see that when $P &lt; 15$, then $Q &gt; 100$ (notice the direction of the inequality is changing; the more expensive a good, the less consumers buy of it)