x^2 -4x < 0
Let us factor x :
Then we have two options:
x < 0 and x-4>0
x<0 and x > 4
This is impossible solution because x can not be < 0 and > 4 at the same time.
x> 0 and x-4 < 0
x>0 and x<4
Then x belongs to the interval (0,4) .......(2)
Then, from (1) and (2) we conclude that:
X belongs to (0,4)
To solve x^2-4x < 0.
X^2-4x < x(x-4) . So the given equation is
x(x-4) < 0.
x=0, x-4 = 0 are the roots.
Therefore as long x x belongs to (0, 4), the expression x and (x-4) have opposite signs. Therefore the product x(x-4) is negative for x in (0,4). Or ) < x <4 for which x(x-4) or x^2-4x < 0.
First of all, let's solve the expression like an equation. For this reason, we'll put:
x^2 -4x = 0
We'll factorize and we'll get:
x(x-4) = 0
We'll put each factor equal to zero.
x = 0
x-4 = 0
x = 4
Now, we'll follow the rule: between the solution of the equation, the expression will have the opposite sign of the coefficient of x^2, that means that x(x-4)<0, outside the solutions, the expression will be positive.
So, the conclusion is that the expression is negative for values of x which are in the interval (0,4).