Given an inequality of the form (x-a), (x-b), how do the factors (x-a), (x-b) split the number line into the following 3 parts?

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.

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

1 Answer

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.