What are those test intevals

In the previous problem, the inequality was $\frac{(x-1)(x-5)}{x+2} < 0$.
The critical points were found by setting the numerator and denominator to zero:

• $x-1 = 0 \implies x=1$
• $x-5 = 0 \implies x=5$
• $x+2 = 0 \implies x=-2$

These critical points ($x=-2, x=1, x=5$) divide the number line into four distinct intervals:

1. $x < -2 \implies (-\infty, -2)$
2. $-2 < x < 1 \implies (-2, 1)$
3. $1 < x < 5 \implies (1, 5)$
4. $x > 5 \implies (5, \infty)$

We then pick a test value from each of these intervals and substitute it into the inequality to determine if the expression is positive or negative in that entire interval. This helps us find which intervals satisfy the original inequality.