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When we encounter a polynomial inequality like \(x^2 - 4x \geq 0\), the first step towards solving it is to factorize the polynomial. Factorizing a polynomial involves breaking it down into simpler, multiplied entities known as factors, that when multiplied out, give us back the original polynomial.

For example, the given polynomial \(x^2 - 4x\) can be factorized by recognizing it as a product of \(x\) and \(x - 4\). In mathematical terms, this process is shown as \(x^2 - 4x = x(x - 4)\). Factorization is essential because it simplifies the problem, allowing us to identify the roots or zeros (solutions) of the equality \(x(x - 4) = 0\), which divides the number line into intervals for further testing. Without this step, proceeding to find the inequality's solution set would be much more complex.

Once the solution of an inequality has been determined, expressing this solution clearly and concisely is where interval notation comes into play. Interval notation is a method of writing subsets of the real number line. A solution set is often composed of all numbers that fall between two specific endpoints, which can either be included in the solution set or not.

For instance, in our inequality \(x^2 - 4x \geq 0\), the solution includes the intervals of \(x < 0\) and \(x > 4\). To express this in interval notation, we use parentheses and brackets. Parentheses, \((\) and \()\), indicate that the endpoint is not included, whereas brackets, \([\) and \(]\), show that the endpoint is included. Thus, our solution set is \((-\text{\infin},0] \cup [4,\text{\infin})\), with brackets around \(0\) and \(4\) to signify that these values satisfy the inequality as well.

To determine which intervals satisfy a polynomial inequality, using test points is an effective method. After factorizing the polynomial and finding the roots to establish intervals, we select a number from each interval—this number is our 'test point'.

We substitute these test points into the original inequality to check if they make the inequality true. If the test point results in a true statement, then all numbers in that interval satisfy the inequality. For example, when testing the interval \(x < 0\) from our inequality \(x^2 - 4x \geq 0\), we might choose -1 as a test point. Since \((-1)^2 - 4(-1) = 1 + 4 > 0\), we confirm that the interval satisfies the inequality. We repeat this process for the other intervals to fully determine the solution set.

Graphing solution sets is a visual way to represent the solutions of inequalities. It involves marking the roots or critical values on a number line and shading the regions that satisfy the inequality.

In our exercise, after identifying the intervals and test points, we graph the solution set by first plotting the roots x = 0 and x = 4 on the number line. We then shade the interval \(-\text{\infin},0\) and \(4,\text{\infin}\) to illustrate the solution. Closed dots on the endpoints indicate that these values are part of the solution set, corresponding to the inclusion of these numbers when the inequality is non-strict (\(\geq\) or \(\leq\)). Graphing not only provides a clear and immediate visual understanding of the solution set but also serves as a final check for the correctness of our solutions.