Logical question for all math lovers?

Question

Find all values of m, so that:

[(x-1)*(x-m)<0] => [(x-2)*(x-m)<0]

holds true for any real x.

Answer 1

this proposition is true when m less than or equal to 1

I did this by thinking of it graphically, these are both parabolas that turn up. The first one has zeros at 1 and m, the second has zeros at 2 and m.

so the first one is negative between 1 and m

the second one is negative between 2 and m

so consider the three cases:

m&lt;1, 1&lt;m&lt;2, and 2&lt;m

if m=1 the vertex is on the x axis so (x-1)(x-m) has no negative values so the  proposition is true because the hypothesis is false for every x.

if m&lt;1 then the interval between 1 and m is a subset of the interval between 2 and m, making the proposition true.

I see this by inspection of the graphs.