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Critical points are crucial when solving inequalities as they indicate where the function's behavior changes. These are the values of \(x\) where the expression equals zero and the inequality can potentially switch from positive to negative or vice versa.
In our example, the inequality was factored into \(x^{2}(x - 3) > 0\). To identify the critical points, set each factor equal to zero and solve:

• For \(x^2 = 0\), we find that \(x = 0\).
• For \(x - 3 = 0\), we find that \(x = 3\).

Thus, our critical points are \(x = 0\) and \(x = 3\). These points help us section the number line into intervals where we can check the inequality's validity.

Factoring is the process of breaking down an expression into its simpler components, or "factors." This step simplifies the process of solving or analyzing equations and inequalities.
In the given inequality \(4x^{3} - 12x^{2} > 0\), the first step was to factor out the greatest common factor, which is \(x^2\). The expression then becomes \(x^{2}(4x - 12) > 0\).
Further simplifying, divide \(4x - 12\) by 4 to get \(x^{2}(x - 3) > 0\).
This makes it easier to find the critical points and evaluate the inequality's sign in different intervals. Factoring is often an essential technique in simplifying and solving expressions both in equations and inequalities.

Once you determine the intervals on the number line where the inequality holds, you can express the solution set using interval notation. This concise form helps communicate the range of \(x\) values that satisfy the inequality.
Based on the critical points identified (\(x = 0\) and \(x = 3\)), test values from each interval to check where \(x^{2}(x - 3) > 0\) holds true. Here, choosing test points such as \(-1\), \(1\), and \(4\), the expression is positive only in the interval where \(x > 3\).
Thus, the solution in interval notation becomes \((3, +\infty)\), meaning that \(x\) must be greater than \(3\) for the inequality to be true. Interval notation provides a clear, efficient way to express solutions to inequalities.