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In the context of polynomial inequalities, 'critical points' are values where the expression equals zero. These are pivotal because they divide the number line into distinct regions or intervals where the inequality might hold true. To find these points, set the polynomial equal to zero and solve for the variable.

For the polynomial inequality \((x-4)(x+2)>0\), equating it to zero means solving \((x-4)(x+2)=0\). This leads to two critical points: \(x = 4\) and \(x = -2\).

Identifying these points provides a foundation for further analysis. They serve as the boundaries between regions on the real number line where the inequality's truth value may differ. Understanding critical points helps in comprehensively analyzing inequalities.

Once the critical points are known, the next step in solving a polynomial inequality involves 'interval testing'. This method determines which intervals satisfy the inequality.

Between the critical points found in the earlier step, you need to divide the number line into segments or intervals. For the inequality \((x-4)(x+2)>0\), these intervals are \(-\infty < x < -2\), \(-2 < x < 4\), and \(4 < x < \infty\).

To test an interval, choose any sample value from within that interval and substitute it into the inequality to see if it holds true. For example, choose \(x = -3\) from the first interval. Substituting \(-3\) gives \(( -3 - 4)( -3 + 2) = -7\), which is not greater than zero. So this interval doesn't satisfy the inequality. Continue this process for other intervals until you determine where the inequality is true.

After identifying which intervals satisfy a polynomial inequality, graphing the solution provides a visual representation of these results. This is where 'inequality solution graphing' comes into play.

To graph the solution for \((x-4)(x+2)>0\), first mark the critical points \(x = -2\) and \(x = 4\) on the number line. Since these values correspond to making the inequality equal zero, they aren't part of the solution set and are typically marked with open circles.

Next, based on interval testing, the solution to the inequality is from \(x = 4\) to \(\infty\), exclusively. On a number line, this interval is represented by a ray or line segment starting just beyond \(x = 4\) and extending in the positive direction toward infinity.

This graphical representation helps in quickly understanding the metric space where the inequality holds, offering an easy-to-interpret method of communicating solutions.