Directions: Determine whether the given is a rational function, a rational equation, a rational
inequality or none of these.
A.
1. 2 + x
x − 1
= 8
2. X > √x + 2
3. f(x) = 6 -
x+3
x
2−5

4. 2x ≥
7
x+ 4
5. 1
2
=
4
x + 9x
3

6. 3x
x +6
+
2
x−3
= 2

Let's analyze each of the given expressions to classify them as a rational function, a rational equation, a rational inequality, or none of these.

### 1. \( \frac{2 + x}{x - 1} = 8 \)
- **Type**: This is a **rational equation**. It involves a fraction where both the numerator and denominator are polynomials, and it's set equal to a constant (8).

### 2. \( x > \sqrt{x + 2} \)
- **Type**: This is **none of these**. It is an inequality, but it is not a rational inequality because the right side involves a square root, not a rational expression.

### 3. \( f(x) = 6 - \frac{x + 3}{x^2 - 5} \)
- **Type**: This is a **rational function**. The expression defines a function \( f(x) \), and it involves a rational expression, which is a quotient of polynomials.

### 4. \( 2x \geq \frac{7}{x + 4} \)
- **Type**: This is a **rational inequality**. It involves an inequality where one side is a rational expression and the other side is a linear polynomial.

### 5. \( \frac{1}{2} = \frac{4}{x + 9x^3} \)
- **Type**: This is a **rational equation**. Both sides of the equation are rational expressions.

### 6. \( \frac{3x}{x + 6} + \frac{2}{x - 3} = 2 \)
- **Type**: This is a **rational equation**. It involves a sum of rational expressions set equal to a constant (2).

### Summary:
1. Rational equation
2. None of these
3. Rational function
4. Rational inequality
5. Rational equation
6. Rational equation

Would you like further details or have any questions? Here are 8 related questions for further exploration:

1. What distinguishes a rational function from a polynomial function?
2. Can you provide an example of a rational inequality that includes two rational expressions?
3. How would you solve a rational equation involving multiple fractions?
4. What is the general method for determining the domain of a rational function?
5. How do you graph a simple rational function?
6. What techniques are used to solve inequalities involving square roots, like in question 2?
7. How do asymptotes relate to rational functions?
8. What are common mistakes to avoid when solving rational equations?

**Tip**: When working with rational functions, always check for domain restrictions, where the denominator equals zero.

Explore rational functions, equations, and inequalities in algebra with detailed explanations and examples. Learn to classify and solve rational expressions, equations, and inequalities effectively. Suitable for Grades 11-12 students.

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