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I encountered this statement on the U of T math department website (under the first example):

In general, if you want to solve an inequality of the form $(x-a)(x-b) > 0$ [...], notice that the factors split the number line into 3 parts: $x < a, a < x < b, x > b$.

I was wondering if anyone could explain why this is true, since I wasn't able to figure it out on my own.

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  • $\begingroup$ Examine the sign of the function $$ f(x) = (x-a)(x-b) $$ on each of these three parts. For simplicity, you can start with $a = 0, b = 1$. $\endgroup$ – avs Apr 19 at 22:57
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Is it because of this?

We know both $(x - a)$ and $(x -b)$ must be either negative or positive, so the possibilities are:

either $x - a > 0$ and $x - b > 0$, or $x - a < 0$ and $x - b < 0$

which is the same as

either $x > a$ and $x > b$, or $x < a$ and $x < b$.

When we map these possibilities on a number line, we end up with the three parts described above.

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