The expression you provided is:

\[
\frac{5x-3}{x-5}
\]

### 1. Simplification
The expression is already in its simplest form since the numerator and denominator have no common factors. Therefore, we cannot simplify it further without changing the expression.

### 2. Properties and Analysis
#### Asymptotes and Discontinuities:
- **Vertical Asymptote**: This occurs where the denominator equals zero. Therefore, for \( x - 5 = 0 \), \( x = 5 \) is a vertical asymptote.
- **Horizontal Asymptote**: Since the degrees of the numerator and denominator are equal (both are first-degree polynomials), the horizontal asymptote can be found by dividing the leading coefficients of the numerator and the denominator. In this case, it is \( \frac{5}{1} = 5 \).
  
#### Domain:
- The domain of this expression is all real numbers except where the denominator is zero. Thus, the domain is:
  
  \[
  x \in \mathbb{R}, \quad x \neq 5
  \]

#### Behavior Near Asymptotes:
- As \( x \) approaches 5 from the left (\( x \to 5^- \)), the function \( \frac{5x-3}{x-5} \) goes to negative infinity.
- As \( x \) approaches 5 from the right (\( x \to 5^+ \)), the function \( \frac{5x-3}{x-5} \) goes to positive infinity.

#### Intercepts:
- **y-intercept**: To find the y-intercept, set \( x = 0 \):

  \[
  y = \frac{5(0)-3}{0-5} = \frac{-3}{-5} = \frac{3}{5}
  \]

  Therefore, the y-intercept is \( \left(0, \frac{3}{5}\right) \).
  
- **x-intercept**: To find the x-intercept, set the numerator equal to zero:

  \[
  5x - 3 = 0 \quad \Rightarrow \quad 5x = 3 \quad \Rightarrow \quad x = \frac{3}{5}
  \]

  Therefore, the x-intercept is \( \left(\frac{3}{5}, 0\right) \).

### Graphical Representation

![Graph of y = (5x-3)/(x-5)](https://i.imgur.com/sP3tboC.png)

### Summary
- **Vertical Asymptote**: \( x = 5 \)
- **Horizontal Asymptote**: \( y = 5 \)
- **Domain**: \( x \in \mathbb{R}, \, x \neq 5 \)
- **x-intercept**: \( \left(\frac{3}{5}, 0\right) \)
- **y-intercept**: \( \left(0, \frac{3}{5}\right) \)

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### 8 Relative Questions
1. What are the characteristics of other types of asymptotes?
2. How do we solve rational equations involving this expression?
3. What are the different methods to simplify rational expressions?
4. How can we determine if a rational function has slant asymptotes?
5. What are common applications of rational expressions in real-world problems?
6. How does the graph of \( \frac{5x-3}{x-5} \) compare to other similar functions?
7. What changes occur in the graph if we alter the numerator or denominator?
8. How can we find the limits of rational functions as \( x \) approaches a certain value?

### Tip
When analyzing rational expressions, always factor the numerator and the denominator first to check for any common factors, as this can simplify the expression and reveal removable discontinuities.

Explore the rational function (5x-3)/(x-5) with detailed analysis of its asymptotes, domain, intercepts, and graphical representation. Learn about vertical and horizontal asymptotes, domain restrictions, intercepts, and graphical behavior. Suitable for high school students and anyone interested in advanced algebra.

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