,Inequalities
It's given the inequality:
x(x - 3) > 0
Note the left side is the product of two expressions: x and x-3.
That product must be greater than zero, or positive.
Recall that the product of two numbers is positive in two different cases:
Both are positive, for example 5*4=20
Both are negative, for example (-5)*(-4) = 20
This means that we can provide two different answers to the inequality, both valid:
1. When x >0 AND x - 3 >0 (both positive), or
2. When x <0 AND x - 3 < 0 (both negative).
The first condition leads to:
x>0 AND x>3. If we intersect these conditions, the solution for this part is x>3.
The second condition gives us:
x<0 AND x<3. The intersection of these conditions gives x<0.
Thus, the final solution is the union (OR) both partial intervals above, i.e.
x>3 OR x<0 --> Option B)