Nafiul,
It's important you know and understand the critical points method for questions such as:
(x - a)(x - b)(x - c)...(x -k) > 0 or < 0
Let me teach you the method here.
for the function f(x) = (x - a)(x - b)(x - c)...(x - k), Critical Points are those points at which f(x) = 0. So a, b, c.. k are the critical points of f(x).
Example 1:
Solve
(x + 1).(x - 2) < 0
Step 1. Find out the critical points
The critical points are -1 and 2 in this case.
Step 2. Plot them on a number line in a proper order.
Step 3: Consider the region above the number line to be positive and below it to be negative. Start drawing a curve starting from the right most - top side as shown below
And then complete the curve in an alternate up-down fashion as shown below
Mark '+' in the region above the number line and '-' in the region below the number line as shown below.
In the '+' region f(x) > 0 and in the '-' region f(x) < 0.
Since the question wants us to solve for f(x) < 0, the values of x for which the curve goes below the number line will be chosen.
The shaded region shown below is our answer.
Answer: -1 < x < 2
Example 2: Lets take the question posted by you.
(x - 3)(x - 2)(x - 1) > 0
Step 1: The critical points are 1, 2 and 3.
Step 2: Mark them on a Number Line.
Step 3: Plot the curve starting from top-right side in an alternate fashion.
Step 4: Mark alternate '+' and '-' and choose the appropriate region
In this case the region in which f(x) > 0 is favourable to us is shown below
Answer: 1 < x < 2 , x > 3
