Intermediate Algebra Study Guide: Complex Fractions & Equivalent Expressions

Q1. Simplify the complex fraction:

$$ \frac{\frac{2}{5-x} + \frac{3}{x-5}}{\frac{2}{x} + \frac{3}{x-5}}$$

Background

Topic: Simplifying Complex Fractions

This question tests your ability to simplify complex fractions, which are fractions where the numerator, denominator, or both contain fractions themselves. The process often involves finding common denominators and combining terms.

Key Terms and Formulas:

• Complex Fraction: A fraction in which the numerator and/or denominator contains one or more fractions.

• Common Denominator: A shared denominator used to combine fractions.

• Simplification Steps: Multiply numerator and denominator by the least common denominator (LCD) to clear the smaller fractions.

Step-by-Step Guidance

1. Identify the denominators in the numerator and denominator: $5-x$, $x-5$, and $x$.

2. Find the least common denominator (LCD) for all fractions involved. Consider $5-x$, $x-5$, and $x$.

3. Rewrite each fraction in the numerator and denominator using the LCD, so you can combine them into single fractions.

4. Once you have a single fraction in the numerator and denominator, set up the division of these two fractions.

5. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Set up this multiplication, but do not compute the final result yet.

Try solving on your own before revealing the answer!

Final Answer:

The simplified form is $\frac{2}{x}$.

After finding the LCD and combining terms, the complex fraction reduces to $\frac{2}{x}$. This shows the importance of careful algebraic manipulation and recognizing equivalent expressions.

Q2. Which of the following are equivalent to $\frac{x+2}{8} \div \frac{9}{y-1}$?

Background

Topic: Equivalent Expressions & Fraction Division

This question tests your understanding of dividing fractions and recognizing equivalent algebraic expressions. It also checks your ability to rewrite division as multiplication by the reciprocal.

Key Terms and Formulas:

• Fraction Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

• Equivalent Expressions: Expressions that have the same value for all values of the variables.

Step-by-Step Guidance

1. Start by rewriting the division as multiplication by the reciprocal: $\frac{x+2}{8} \div \frac{9}{y-1} = \frac{x+2}{8} \times \frac{y-1}{9}$.

2. Multiply the numerators and denominators: $\frac{(x+2)(y-1)}{8 \times 9}$.

3. Check the answer choices to see which ones match this form or are equivalent after simplification.

4. Be careful to compare each option, looking for equivalent forms (e.g., $\frac{x+2}{8} \times \frac{y-1}{9}$ or $\frac{(x+2)(y-1)}{72}$).

Try solving on your own before revealing the answer!

Final Answer:

The equivalent expressions are:

• A. $\frac{x+2}{8} \times \frac{y-1}{9}$

• C. $\frac{(x+2)(y-1)}{72}$

Both expressions represent the same value as the original division, using the rule for dividing fractions.