91Ó°ÊÓ

Identifying critical points is a key step in solving inequalities, like our given example \((x-2)(x+3)>0\). Critical points are the values of \(x\) where each factor in the inequality equals zero.
For the equation \(x-2=0\), solving gives us \(x=2\). Similarly, the equation \(x+3=0\) results in \(x=-3\).
These critical points are vital because they help us divide the number line into intervals. Each interval will show us whether the entire inequality is true or false within that section.
The intervals for this problem will be:

• \((-\infty, -3)\)
• \((-3, 2)\)
• \((2, \infty)\)

This breakdown is the foundation for the next step, interval testing.

Once we have our intervals from the critical points, interval testing helps us determine where the inequality holds true. We select a test point from each interval and substitute it into the inequality expression \((x-2)(x+3)\).
In the interval \((-\infty, -3)\), choosing \(x = -4\) results in \((-4-2)(-4+3) = (-6)(-1) = 6\). Since 6 is greater than 0, the inequality is true here.
For \((-3, 2)\), selecting \(x = 0\) gives \((0-2)(0+3) = (-2)(3) = -6\), indicating a false inequality because -6 is less than 0.
Finally, in the interval \((2, \infty)\), choosing \(x=3\) results in \((3-2)(3+3) = (1)(6) = 6\). Again, the inequality is true because 6 is positive.

• The inequality holds for intervals \((-\infty, -3)\)
• and \((2, \infty)\)

This step confirms the parts of the number line where the original inequality is satisfied.

Graphing the solution on a number line visually shows where the inequality is true. Begin by marking the critical points \(x=-3\) and \(x=2\) with open circles. These are open because the inequality \(>0\) excludes the critical points themselves.
To represent the solution intervals from the previous step, draw a line starting from \((-\infty, -3)\) and another from \((2, \infty)\). Do not include the sections between \(-3\) and \(2\) since the solution is false there.
This visual aid helps confirm the accurate regions on the number line where \((x-2)(x+3)>0\). It ensures that our testing and calculations correctly identified positive intervals, and it provides a quick, accessible way to verify solutions.