Question

which graph shows the solution set of \\(\frac{x - 1}{x - 3}<0\\)?

Explanation:

Step1: Analyze the rational - inequality

For the inequality $\frac{x - 1}{x - 3}<0$, we need to find the values of $x$ that make the fraction negative. The critical points are found by setting the numerator and denominator equal to zero.
Set $x−1 = 0$, then $x = 1$; set $x−3=0$, then $x = 3$.

Step2: Test intervals

We have three intervals to test: $x<1$, $1 3$.
For $x<1$, let's take $x = 0$. Then $\frac{0 - 1}{0 - 3}=\frac{-1}{-3}=\frac{1}{3}>0$.
For $1For $x>3$, let's take $x = 4$. Then $\frac{4 - 1}{4 - 3}=\frac{3}{1}=3>0$.
So the solution of the inequality $\frac{x - 1}{x - 3}<0$ is $1

Answer:

The graph with open - circles at $x = 1$ and $x = 3$ and a line segment between them on the number - line represents the solution set of the inequality $\frac{x - 1}{x - 3}<0$.