Solve 3x(x-4)<x(x-1)

$$3 x ( x - 4 ) < x ( x - 1 )$$

Solve for x

$$x\in \left(0,\frac{11}{2}\right)$$

Solution Steps

Use the distributive property to multiply 3x by x-4.

$$3x^{2}-12x<x\left(x-1\right)$$

Use the distributive property to multiply x by x-1.

$$3x^{2}-12x<x^{2}-x$$

Subtract x^{2} from both sides.

$$3x^{2}-12x-x^{2}<-x$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}-12x<-x$$

Add x to both sides.

$$2x^{2}-12x+x<0$$

Combine -12x and x to get -11x.

$$2x^{2}-11x<0$$

Factor out x.

$$x\left(2x-11\right)<0$$

For the product to be negative, x and x-\frac{11}{2} have to be of the opposite signs. Consider the case when x is positive and x-\frac{11}{2} is negative.

$$x>0 x-\frac{11}{2}<0$$

The solution satisfying both inequalities is x\in \left(0,\frac{11}{2}\right).

$$x\in \left(0,\frac{11}{2}\right)$$

Consider the case when x-\frac{11}{2} is positive and x is negative.

$$x-\frac{11}{2}>0 x<0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(0,\frac{11}{2}\right)$$