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**Part I: Identifying expressions**

You are tasked with identifying whether the following expressions are rational expressions (REx), rational equations (REq), rational inequalities (RI), or rational functions (RF).

### 1. \(\frac{7}{ab}\)
- **Answer**: Rational expression (REx). 
This is a fraction where the numerator and denominator are polynomials.

### 2. \(\frac{5}{x-2} = y\)
- **Answer**: Rational equation (REq). 
It's an equation involving a rational expression.

### 3. \(\frac{7-3x}{2} = \frac{5}{8}\)
- **Answer**: Rational equation (REq). 
Two rational expressions are equated, so it's a rational equation.

### 4. \(\frac{7}{x} < \frac{3}{4}\)
- **Answer**: Rational inequality (RI). 
It involves a rational expression and inequality comparison.

### 5. \(g(x) = \frac{x-1}{6}\)
- **Answer**: Rational function (RF). 
This is a function where the output is a rational expression in terms of \(x\).

### 6. \(\frac{x-5}{x+1} = \frac{1}{2}\)
- **Answer**: Rational equation (REq). 
An equation involving rational expressions.

### 7. \(\frac{x+5}{x-2} \geq 2\)
- **Answer**: Rational inequality (RI). 
Inequality involving rational expressions.

### 8. \(f(x) = \frac{5}{8x}\)
- **Answer**: Rational function (RF). 
This is a rational function in terms of \(x\).

### 9. \(\frac{7}{2x-1} \geq \frac{x}{x}\)
- **Answer**: Rational inequality (RI). 
This is a comparison of rational expressions using inequality.

### 10. \(4 - \frac{x}{5} = \frac{x-1}{6}\)
- **Answer**: Rational equation (REq). 
Two rational expressions equated, making it a rational equation.

---

**Part II: Solving Rational Equations**

### 1. \(\frac{2}{x} + \frac{3}{2x} = \frac{1}{5}\)
We'll solve this equation for \(x\).

Multiply everything by the least common denominator (LCD) of \(10x\):
\[
10x\left(\frac{2}{x} + \frac{3}{2x} = \frac{1}{5}\right)
\]
\[
20 + 15 = 2x
\]
\[
35 = 2x
\]
\[
x = \frac{35}{2}
\]
So, the solution is \(x = \frac{35}{2}\).

### 2. \(\frac{3}{x-3} = \frac{3}{x+1}\)

Cross-multiply:
\[
3(x+1) = 3(x-3)
\]
\[
3x + 3 = 3x - 9
\]
\[
3 \neq -9
\]
There is **no solution** to this equation since it results in a contradiction.

### 3. \(\frac{x}{x+2} + \frac{1}{x-2} = \frac{8}{x^2-4}\)

Factor the denominator on the right-hand side:
\[
\frac{x}{x+2} + \frac{1}{x-2} = \frac{8}{(x+2)(x-2)}
\]

Combine the left-hand side over a common denominator:
\[
\frac{x(x-2) + 1(x+2)}{(x+2)(x-2)} = \frac{8}{(x+2)(x-2)}
\]
Simplify the numerator:
\[
\frac{x^2 - 2x + x + 2}{(x+2)(x-2)} = \frac{8}{(x+2)(x-2)}
\]
\[
\frac{x^2 - x + 2}{(x+2)(x-2)} = \frac{8}{(x+2)(x-2)}
\]
Since the denominators are the same, equate the numerators:
\[
x^2 - x + 2 = 8
\]
\[
x^2 - x - 6 = 0
\]
Factor the quadratic:
\[
(x - 3)(x + 2) = 0
\]
Thus, \(x = 3\) or \(x = -2\).

However, \(x = -2\) is not valid because it would make the denominator zero in the original equation. So, the solution is:
\[
x = 3
\]

---

Do you want further clarifications or have any questions?

Here are five questions related to this topic:
1. What defines a rational expression?
2. How do we identify rational inequalities?
3. What makes a function a rational function?
4. Why do some rational equations have no solution?
5. How do we solve more complex rational equations involving multiple terms?

**Tip**: When solving rational equations, always check for excluded values that would make the denominator zero, as these lead to undefined expressions.

Identify whether the following expressions are rational expressions, rational equations, rational inequality or rational functions. Write REx for rational expression, REq for rational equation, RI for rational inequality, or RF for rational function.

Learn how to identify rational expressions, equations, inequalities, and functions in algebra. This problem explains the different types of rational forms and includes step-by-step solutions. Suitable for students in Grades 9-11.

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