Q1. Subtract and simplify: \( \frac{-5a}{a^2 - a - 2} - \frac{10}{a^2 - a - 2} \)

Background

Topic: Subtracting Rational Expressions

This question tests your ability to subtract rational expressions with like denominators and simplify the result.

Key Terms and Formulas:

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

• Like Denominators: When denominators are the same, you can combine numerators directly.

Step-by-Step Guidance

1. Notice both fractions have the same denominator: \( a^2 - a - 2 \).

2. Combine the numerators: \( -5a - 10 \), keeping the common denominator.

3. Write the combined expression: \( \frac{-5a - 10}{a^2 - a - 2} \).

4. Factor the numerator and denominator if possible to simplify further.

Try solving on your own before revealing the answer!

Q2. Add and simplify: \( \frac{y + 8}{y - 2} + \frac{y + 9}{y - 2} \)

Background

Topic: Adding Rational Expressions

This question tests your ability to add rational expressions with like denominators and simplify the result.

Key Terms and Formulas:

• Like Denominators: Add numerators directly when denominators are the same.

Step-by-Step Guidance

1. Check that both denominators are \( y - 2 \).

2. Add the numerators: \( (y + 8) + (y + 9) \).

3. Combine like terms in the numerator.

4. Write the result over the common denominator and check if it can be simplified further.

Try solving on your own before revealing the answer!

Q3. Subtract and simplify: \( \frac{-6x^2}{x + 4} - \frac{16 + 5x^2}{x + 4} \)

Background

Topic: Subtracting Rational Expressions

This question tests your ability to subtract rational expressions with like denominators and simplify the result.

Key Terms and Formulas:

• Like Denominators: Subtract numerators directly when denominators are the same.

Step-by-Step Guidance

1. Both denominators are \( x + 4 \).

2. Subtract the numerators: \( -6x^2 - (16 + 5x^2) \).

3. Simplify the numerator by distributing the negative sign and combining like terms.

4. Write the result over the common denominator and check for further simplification.

Try solving on your own before revealing the answer!

Q4. Find the LCD for: \( \frac{1}{10x}, \frac{5}{2x + 2} \)

Background

Topic: Least Common Denominator (LCD) of Rational Expressions

This question tests your ability to find the least common denominator for two rational expressions.

Key Terms and Formulas:

• LCD: The smallest expression that is a multiple of each denominator.

• Factor denominators completely to find the LCD.

Step-by-Step Guidance

1. Factor each denominator: \( 10x \) and \( 2x + 2 = 2(x + 1) \).

2. List all unique factors from both denominators.

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

Try solving on your own before revealing the answer!

Q5. Find the LCD for: \( \frac{2}{x + 2}, \frac{9}{x - 4} \)

Background

Topic: Least Common Denominator (LCD) of Rational Expressions

This question tests your ability to find the LCD for two rational expressions with linear denominators.

Key Terms and Formulas:

• LCD: The product of all unique factors in the denominators.

Step-by-Step Guidance

1. Identify the denominators: \( x + 2 \) and \( x - 4 \).

2. Since there are no common factors, the LCD is their product.

3. Write the LCD in factored form.

Try solving on your own before revealing the answer!

Q6. Find the LCD for: \( \frac{19}{4x + 8}, \frac{15}{2x^2 + 8x + 8} \)

Background

Topic: Least Common Denominator (LCD) of Rational Expressions

This question tests your ability to factor denominators and find the LCD for two rational expressions.

Key Terms and Formulas:

• Factor denominators completely before finding the LCD.

Step-by-Step Guidance

1. Factor \( 4x + 8 \) and \( 2x^2 + 8x + 8 \).

2. List all unique factors from both denominators.

3. Multiply each unique factor together to form the LCD, in expanded or factored form.

Try solving on your own before revealing the answer!

Q7. Rewrite \( \frac{5}{8x + 24} \) as an equivalent rational expression with denominator \( 8z(x + 3) \)

Background

Topic: Equivalent Rational Expressions

This question tests your ability to rewrite a rational expression with a new denominator by finding an appropriate multiplier.

Key Terms and Formulas:

• To rewrite with a new denominator, multiply numerator and denominator by the necessary factor.

Step-by-Step Guidance

1. Factor the original denominator: \( 8x + 24 = 8(x + 3) \).

2. Compare the factored denominator to the new denominator \( 8z(x + 3) \).

3. Determine what factor is missing and multiply numerator and denominator by that factor.

Try solving on your own before revealing the answer!

Q8. Rewrite \( \frac{7t + 9}{15t + 24} \) as an equivalent rational expression with denominator \( 3s(5t + 8) \)

Background

Topic: Equivalent Rational Expressions

This question tests your ability to rewrite a rational expression with a new denominator by finding the appropriate factor.

Key Terms and Formulas:

• Factor the original denominator and compare to the new denominator.

Step-by-Step Guidance

1. Factor \( 15t + 24 \) if possible.

2. Compare to \( 3s(5t + 8) \) to see what is missing.

3. Multiply numerator and denominator by the missing factor to get the new denominator.

Try solving on your own before revealing the answer!

Q9. Rewrite \( \frac{x}{x^3 + 7x^2 + 6x} \) as an equivalent rational expression with denominator \( x(x + 7)(x + 1)(x + 6) \)

Background

Topic: Equivalent Rational Expressions

This question tests your ability to factor polynomials and rewrite rational expressions with a specified denominator.

Key Terms and Formulas:

• Factor the denominator completely.

• Multiply numerator and denominator by any missing factors to match the new denominator.

Step-by-Step Guidance

1. Factor \( x^3 + 7x^2 + 6x \) completely.

2. Compare the factored form to the new denominator.

3. Multiply numerator and denominator by any missing factors to obtain the new denominator.

Try solving on your own before revealing the answer!

Q10. Add and simplify: \( \frac{49}{7x} + \frac{36}{6x} \)

Background

Topic: Adding Rational Expressions with Unlike Denominators

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

Key Terms and Formulas:

• Find the least common denominator (LCD) for the denominators.

• Rewrite each fraction with the LCD, then add numerators.

Step-by-Step Guidance

1. Factor denominators: \( 7x \) and \( 6x \).

2. Find the LCD of \( 7x \) and \( 6x \).

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

4. Add the numerators and simplify if possible.

Try solving on your own before revealing the answer!

Q11. Subtract and simplify: \( \frac{-4}{x + 3} - \frac{3x}{x^2 - 9} \)

Background

Topic: Subtracting Rational Expressions with Unlike Denominators

This question tests your ability to find a common denominator, rewrite each fraction, and subtract.

Key Terms and Formulas:

• Factor denominators to find the LCD.

• Rewrite each fraction with the LCD, then subtract numerators.

Step-by-Step Guidance

1. Factor \( x^2 - 9 = (x + 3)(x - 3) \).

2. The LCD is \( (x + 3)(x - 3) \).

3. Rewrite \( \frac{-4}{x + 3} \) with the LCD as denominator.

4. Subtract the numerators and combine like terms.

Try solving on your own before revealing the answer!

Q12. Add and simplify: \( \frac{3}{c - 6} + \frac{3}{6 - c} \)

Background

Topic: Adding Rational Expressions with Opposite Denominators

This question tests your ability to recognize and manipulate denominators that are negatives of each other.

Key Terms and Formulas:

• \( 6 - c = -(c - 6) \)

• Rewrite one denominator to match the other, adjusting the numerator's sign as needed.

Step-by-Step Guidance

1. Recognize that \( 6 - c = -(c - 6) \).

2. Rewrite \( \frac{3}{6 - c} \) as \( -\frac{3}{c - 6} \).

3. Add the numerators over the common denominator.

4. Simplify the result.

Try solving on your own before revealing the answer!

Q13. Add and simplify: \( 7 + \frac{5}{x + 9} \)

Background

Topic: Adding a Whole Number and a Rational Expression

This question tests your ability to write a whole number as a rational expression and add it to another rational expression.

Key Terms and Formulas:

• Express the whole number with the same denominator as the rational expression.

Step-by-Step Guidance

1. Rewrite 7 as \( \frac{7(x + 9)}{x + 9} \).

2. Add the numerators over the common denominator.

3. Simplify the numerator if possible.

Try solving on your own before revealing the answer!

Q14. Subtract and simplify: \( \frac{x}{x^2 - 4} - \frac{1}{x^2 - 4x + 4} \)

Background

Topic: Subtracting Rational Expressions with Unlike Denominators

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

Key Terms and Formulas:

• Factor denominators: \( x^2 - 4 = (x + 2)(x - 2) \), \( x^2 - 4x + 4 = (x - 2)^2 \).

• Find the LCD and rewrite each fraction with the LCD.

Step-by-Step Guidance

1. Factor both denominators.

2. Find the LCD: \( (x + 2)(x - 2)^2 \).

3. Rewrite each fraction with the LCD as denominator.

4. Subtract the numerators and simplify.

Try solving on your own before revealing the answer!

Q15. Add and simplify: \( \frac{4}{3 - x} + \frac{x}{4x - 12} \)

Background

Topic: Adding Rational Expressions with Unlike Denominators

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

Key Terms and Formulas:

• Factor denominators: \( 4x - 12 = 4(x - 3) \).

• Find the LCD and rewrite each fraction with the LCD.

Step-by-Step Guidance

1. Factor \( 4x - 12 \) and consider how to rewrite \( 3 - x \) in terms of \( x - 3 \).

2. Find the LCD for both denominators.

3. Rewrite each fraction with the LCD as denominator.

4. Add the numerators and simplify if possible.

Try solving on your own before revealing the answer!