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If I have demand functions

For $P<15$:

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

For $P>15$: $$ Q(P) = 400-20P $$

If I invert them, I get the price functions

For $Q<100$: $$ P(Q)=20 - (1/20)Q $$

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

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 $P < 15$ can change to $Q < 100$.

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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 $P < 15$, then $Q > 100$ (notice the direction of the inequality is changing; the more expensive a good, the less consumers buy of it)

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