Question

solve (x-1)(x-2)/x-3 >_ 0

please help me so this step by step

Answer 1

you want to know where it is positive, which his a synonym for \(>0\)
it changes sign at the zero of each factor, namely at \(1,2,3\) so you need to consider four intervals
\[(-\infty,1), (1,2),(2,3),(3,\infty)\]

Answer 2

test the expression at any point in any one of these intervals, it doesn't matter what point you pick or what interval you are in. you could pick say \(x=0\) or \(x=4\) or anything
i will pick \(x=0\)

Answer 3

if we replace \(x\) by \(0\) we get
\[\frac{-1\times (-2)}{-3}=-\frac{2}{3}\] which is negative.

Answer 4

than means it is negative on the interval \((-\infty,1)\) then positive on the interval \((1,2)\) then negative on the interval \((2,3)\) and finally positive on the interval \((3,\infty)\)

Answer 5

you want to know where it is positive, but notice you know where it is positive, negative, zero, and undefined. your solution is
\[(1,2)\cup (3,\infty)\]

Answer 6

Thank you!

Answer 7

Thank you!