Question

Solve the inequality algebraically

$x^3-4x^2>0$

Short answer

Required solution set is $(4,\infty )$.

Step-by-step solution

Step 1. Given information  

We are given an inequality,

$x^3-4x^2>0$

Step 2.Finding the zeros   

The zeros are given by,

$$\begin{gathered} x^3-4x^2=0 \\ {x}^2(x-4)=0 \\ x=0 \\ x-4=0 \\ x=4 \end{gathered}$$

So, the intervals will be,

$(-\infty ,0),(0,4)\text{and}(4,\infty )$

Step 4. Selecting a test number in each interval

The table is as follows,

Interval	$(-\infty ,0)$	$(0,4)$	$(4,\infty )$
Chosen Number	$-1$	$1$	$5$
Value of Function	$f(-1)=-5$	$f\left(1\right)=-3$	$f\left(5\right)=25$
Check	Negative	Negative	Positive

Hence the solution set is $(4,\infty )$.