x^2 -4x < 0

Let us factor x :

x(x-4) <0

Then we have two options:

option 1:

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.

Option 2:

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).**