Intermediate Algebra Study Guide: Adding, Subtracting, and Finding LCD of Rational Expressions

Q1. Add the following rational expressions:

a. $\frac{3}{8} + \frac{7}{8}$ b. $\frac{4}{3} + \frac{1}{3}$

Background

Topic: Adding Rational Expressions

This question tests your ability to add rational expressions (fractions) with common denominators.

Key Terms and Formulas:

• Rational Expression: A fraction where the numerator and denominator are polynomials.

• Common Denominator: When denominators are the same, you can add numerators directly.

Step-by-Step Guidance

1. Check if the denominators are the same for each pair of fractions.

2. If denominators are the same, add the numerators together and keep the denominator unchanged.

3. Write the sum as a single fraction.

4. Consider if the resulting fraction can be simplified.

Try solving on your own before revealing the answer!

Final Answer:

a. $\frac{3}{8} + \frac{7}{8} = \frac{10}{8} = \frac{5}{4}$ b. $\frac{4}{3} + \frac{1}{3} = \frac{5}{3}$

Since the denominators are the same, you simply add the numerators and simplify if possible.

Q2. Subtract the following rational expressions:

a. $\frac{5}{9} - \frac{2}{9}$ b. $\frac{7}{4} - \frac{3}{4}$

Background

Topic: Subtracting Rational Expressions

This question tests your ability to subtract rational expressions with common denominators.

Key Terms and Formulas:

• Subtracting Fractions: If denominators are the same, subtract the numerators.

Step-by-Step Guidance

1. Check if the denominators are the same for each pair of fractions.

2. If denominators are the same, subtract the second numerator from the first and keep the denominator unchanged.

3. Write the difference as a single fraction.

4. Consider if the resulting fraction can be simplified.

Try solving on your own before revealing the answer!

Final Answer:

a. $\frac{5}{9} - \frac{2}{9} = \frac{3}{9} = \frac{1}{3}$ b. $\frac{7}{4} - \frac{3}{4} = \frac{4}{4} = 1$

Subtract the numerators and simplify the result if possible.

Q3. Subtract the following rational expressions:

a. $\frac{11}{12} - \frac{5}{12}$ b. $\frac{8}{7} - \frac{2}{7}$

Background

Topic: Subtracting Rational Expressions

This question tests your ability to subtract rational expressions with common denominators.

Key Terms and Formulas:

• Subtracting Fractions: If denominators are the same, subtract the numerators.

Step-by-Step Guidance

1. Check if the denominators are the same for each pair of fractions.

2. If denominators are the same, subtract the second numerator from the first and keep the denominator unchanged.

3. Write the difference as a single fraction.

4. Consider if the resulting fraction can be simplified.

Try solving on your own before revealing the answer!

Final Answer:

a. $\frac{11}{12} - \frac{5}{12} = \frac{6}{12} = \frac{1}{2}$ b. $\frac{8}{7} - \frac{2}{7} = \frac{6}{7}$

Subtract the numerators and simplify the result if possible.

Q4. Find the Least Common Denominator (LCD) of the following rational expressions:

a. $\frac{1}{x^2}$ and $\frac{1}{x}$ b. $\frac{1}{x+2}$ and $\frac{1}{x+3}$

Background

Topic: Least Common Denominator (LCD)

This question tests your ability to identify the LCD for rational expressions with different denominators.

Key Terms and Formulas:

• LCD: The smallest expression that both denominators divide into.

Step-by-Step Guidance

1. Factor each denominator if possible.

2. Identify all unique factors from each denominator.

3. Multiply each unique factor together to form the LCD.

4. Write the LCD as a single expression.

Try solving on your own before revealing the answer!

Final Answer:

a. LCD is $x^2$ b. LCD is $(x+2)(x+3)$

The LCD must include all factors from each denominator.

Q5. Find the LCD of the following rational expressions:

a. $\frac{1}{x^2-1}$ and $\frac{1}{x+1}$ b. $\frac{1}{x^2-4}$ and $\frac{1}{x-2}$

Background

Topic: LCD with Factored Denominators

This question tests your ability to factor denominators and find the LCD.

Key Terms and Formulas:

• Factoring: $x^2-1 = (x+1)(x-1)$, $x^2-4 = (x+2)(x-2)$

• LCD: Includes all unique factors from each denominator.

Step-by-Step Guidance

1. Factor each denominator completely.

2. List all unique factors from each denominator.

3. Multiply each unique factor together to form the LCD.

4. Write the LCD as a single expression.

Try solving on your own before revealing the answer!

Final Answer:

a. LCD is $(x+1)(x-1)$ b. LCD is $(x+2)(x-2)$

Factor denominators and include all unique factors in the LCD.

Q6. Find the LCD of the following rational expressions:

a. $\frac{1}{x^2-5x+6}$ and $\frac{1}{x-2}$ b. $\frac{1}{x^2-4x+4}$ and $\frac{1}{x-2}$

Background

Topic: LCD with Quadratic Denominators

This question tests your ability to factor quadratics and find the LCD.

Key Terms and Formulas:

• Factoring: $x^2-5x+6 = (x-2)(x-3)$, $x^2-4x+4 = (x-2)^2$

• LCD: Includes all unique factors from each denominator.

Step-by-Step Guidance

1. Factor each denominator completely.

2. List all unique factors from each denominator.

3. Multiply each unique factor together to form the LCD.

4. Write the LCD as a single expression.

Try solving on your own before revealing the answer!

Final Answer:

a. LCD is $(x-2)(x-3)$ b. LCD is $(x-2)^2$

Factor quadratics and include all unique factors in the LCD.

Q7. Find the LCD of the following rational expressions:

a. $\frac{1}{x^2-6x+9}$ and $\frac{1}{x-3}$ b. $\frac{1}{x^2-9}$ and $\frac{1}{x+3}$

Background

Topic: LCD with Perfect Square and Difference of Squares

This question tests your ability to factor perfect squares and differences of squares to find the LCD.

Key Terms and Formulas:

• Factoring: $x^2-6x+9 = (x-3)^2$, $x^2-9 = (x+3)(x-3)$

• LCD: Includes all unique factors from each denominator.

Step-by-Step Guidance

1. Factor each denominator completely.

2. List all unique factors from each denominator.

3. Multiply each unique factor together to form the LCD.

4. Write the LCD as a single expression.

Try solving on your own before revealing the answer!

Final Answer:

a. LCD is $(x-3)^2$ b. LCD is $(x+3)(x-3)$

Factor denominators and include all unique factors in the LCD.

Q8. Add the following rational expressions:

$\frac{9}{5} + \frac{14}{7}$

Background

Topic: Adding Rational Expressions with Different Denominators

This question tests your ability to find a common denominator and add rational expressions.

Key Terms and Formulas:

• Common Denominator: The least common multiple of the denominators.

• Rewrite each fraction with the common denominator before adding.

Step-by-Step Guidance

1. Find the least common denominator (LCD) for 5 and 7.

2. Rewrite each fraction with the LCD as the denominator.

3. Add the numerators together.

4. Write the sum as a single fraction.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{9}{5} + \frac{14}{7} = \frac{133}{35}$

Rewrite each fraction with the LCD (35), then add the numerators.

Q9. Subtract the following rational expressions:

$\frac{z^2}{z+7} - \frac{49}{z+7}$

Background

Topic: Subtracting Rational Expressions with Common Denominator

This question tests your ability to subtract rational expressions with the same denominator and simplify.

Key Terms and Formulas:

• Subtracting Fractions: If denominators are the same, subtract the numerators.

• Simplify: Factor and reduce if possible.

Step-by-Step Guidance

1. Check if the denominators are the same (they are: $z+7$).

2. Subtract the numerators: $z^2 - 49$.

3. Write the difference as a single fraction: $\frac{z^2 - 49}{z+7}$.

4. Factor the numerator if possible.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{z^2}{z+7} - \frac{49}{z+7} = \frac{z^2 - 49}{z+7} = \frac{(z+7)(z-7)}{z+7} = z-7$ (for $z \neq -7$)

Subtract the numerators, factor, and simplify.

Q10. Find the sum of the following rational expressions:

$\frac{1}{7w} + \frac{8}{5w}$

Background

Topic: Adding Rational Expressions with Different Denominators

This question tests your ability to find a common denominator and add rational expressions.

Key Terms and Formulas:

• LCD: The least common multiple of $7w$ and $5w$.

• Rewrite each fraction with the LCD as the denominator before adding.

Step-by-Step Guidance

1. Find the LCD for $7w$ and $5w$.

2. Rewrite each fraction with the LCD as the denominator.

3. Add the numerators together.

4. Write the sum as a single fraction.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{1}{7w} + \frac{8}{5w} = \frac{5 + 56}{35w} = \frac{61}{35w}$

Rewrite each fraction with the LCD ($35w$), then add the numerators.

Q11. Subtract the following rational expressions:

$\frac{y+8}{y^2-4y+3} - \frac{2}{y^2-1}$

Background

Topic: Subtracting Rational Expressions with Different Denominators

This question tests your ability to factor denominators, find the LCD, and subtract rational expressions.

Key Terms and Formulas:

• Factoring: $y^2-4y+3 = (y-1)(y-3)$, $y^2-1 = (y-1)(y+1)$

• LCD: The least common multiple of the denominators.

Step-by-Step Guidance

1. Factor each denominator completely.

2. Find the LCD by including all unique factors from each denominator.

3. Rewrite each fraction with the LCD as the denominator.

4. Subtract the numerators and write the result as a single fraction.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{y+8}{y^2-4y+3} - \frac{2}{y^2-1} = \frac{(y+8)(y+1) - 2(y-1)(y-3)}{(y-1)(y-3)(y+1)}$

Factor denominators, find the LCD, rewrite each fraction, and subtract the numerators.