Question

state the solution set to the inequality by reading the graph to the right.
\\(\frac{x - 2}{x - 6}>0\\)
the solution set is \\(\square\\).
(type your answer in interval notation.)

Explanation:

Step1: Find the critical points

Set the numerator and denominator equal to zero. For $x - 2=0$, $x = 2$; for $x - 6=0$, $x = 6$. These are the points where the function $y=\frac{x - 2}{x - 6}$ is zero or undefined.

Step2: Test the intervals

We have three intervals to test: $(-\infty,2)$, $(2,6)$ and $(6,\infty)$.
For the interval $(-\infty,2)$, let's choose $x = 0$. Then $\frac{0 - 2}{0 - 6}=\frac{-2}{-6}=\frac{1}{3}>0$.
For the interval $(2,6)$, let's choose $x = 4$. Then $\frac{4 - 2}{4 - 6}=\frac{2}{-2}=-1<0$.
For the interval $(6,\infty)$, let's choose $x = 7$. Then $\frac{7 - 2}{7 - 6}=\frac{5}{1}=5>0$.

Answer:

$(-\infty,2)\cup(6,\infty)$