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To solve these problems, let's address each one in section B by determining the values of \( x \) that would make each expression undefined. A rational expression is undefined when its denominator equals zero.

### B. Determine the value of \( x \) that will make each rational expression undefined

1. **\( \frac{2}{x + 7} \)**  
   The expression is undefined when the denominator is zero:
   \[
   x + 7 = 0 \implies x = -7
   \]

2. **\( \frac{2}{4x - 3} \)**  
   The expression is undefined when:
   \[
   4x - 3 = 0 \implies x = \frac{3}{4}
   \]

3. **\( \frac{4}{x} \)**  
   The expression is undefined when:
   \[
   x = 0
   \]

4. **\( \frac{x + 1}{x + 5} \)**  
   The expression is undefined when:
   \[
   x + 5 = 0 \implies x = -5
   \]

5. **\( \frac{6 - x^2}{x^3} \)**  
   The expression is undefined when:
   \[
   x^3 = 0 \implies x = 0
   \]

### C. Simplify the Rational Expression

1. **\( \frac{12a^5b^4}{18ab^2} \)**  
   Simplify by factoring out the common terms in the numerator and denominator:
   - Factor out \( 6ab^2 \) from both the numerator and denominator:
     \[
     \frac{12a^5b^4}{18ab^2} = \frac{6 \cdot 2a^5b^4}{6 \cdot 3ab^2} = \frac{2a^{5-1}b^{4-2}}{3} = \frac{2a^4b^2}{3}
     \]
   Final simplified expression:
   \[
   \frac{2a^4b^2}{3}
   \]

Would you like further details on any of these steps?

### Follow-up Questions

1. What values of \( x \) would make other rational expressions undefined?
2. How can you simplify more complex rational expressions with multiple variables?
3. What is the importance of simplifying rational expressions in mathematics?
4. How do you determine if a rational function has any asymptotes?
5. What steps are involved in factoring the numerator and denominator in complex rational expressions?

### Tip

When simplifying rational expressions, always factor both the numerator and the denominator to identify and cancel common terms effectively.

Determine the values of x that make each rational expression in the list undefined and simplify a given rational expression.

Learn how to determine the values of x that make rational expressions undefined and simplify a complex rational expression. This guide covers key algebraic concepts for students in Grades 9-11.

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