Intermediate Algebra Unit 3 Review – Step-by-Step Guidance

Q1. The denominators must be the same before performing the operations ______. (Select all that apply.)

Background

Topic: Operations with Rational Expressions

This question tests your understanding of which operations with fractions or rational expressions require a common denominator.

Key Terms:

• Denominator: The bottom part of a fraction.

• Common Denominator: A shared multiple of the denominators of two or more fractions.

Step-by-Step Guidance

1. Recall that when adding or subtracting fractions, you need a common denominator to combine the numerators correctly.

2. Think about multiplication and division: do you need to adjust denominators before performing these operations?

3. Review the properties of each operation with rational expressions to determine which require common denominators.

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Q2. These operations are commutative (order doesn't matter). (Select all that apply.)

Background

Topic: Properties of Operations

This question is about the commutative property, which states that changing the order of the numbers does not change the result for certain operations.

Key Terms:

• Commutative Property: An operation is commutative if $a \ast b = b \ast a$ for all numbers $a$ and $b$.

• Operations: Addition, subtraction, multiplication, division.

Step-by-Step Guidance

1. Recall which operations (addition, subtraction, multiplication, division) are commutative for real numbers.

2. Test each operation with sample numbers to see if changing the order affects the result.

3. Mark all that apply based on your findings.

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Q3. Write the following expression with positive exponents: $y^{-22}$

Background

Topic: Laws of Exponents

This question tests your ability to rewrite expressions with negative exponents as equivalent expressions with only positive exponents.

Key Terms and Formulas:

• Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ for $a \neq 0$.

Step-by-Step Guidance

1. Identify the base and the exponent in the expression $y^{-22}$.

2. Apply the negative exponent rule to rewrite the expression with a positive exponent.

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Q4. Solve the equation: $\frac{x-2}{x+5} = \frac{14}{x+5}$

Background

Topic: Solving Rational Equations

This question asks you to solve an equation involving rational expressions with like denominators.

Key Terms and Formulas:

• Rational Equation: An equation containing one or more rational expressions.

Step-by-Step Guidance

1. Notice that both sides of the equation have the same denominator ($x+5$).

2. Since the denominators are equal and not zero, set the numerators equal to each other: $x-2 = 14$.

3. Solve for $x$ by isolating the variable.

4. Check that your solution does not make the denominator zero.

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Q5. Solve the equation: $\frac{2}{x+3} + \frac{1}{x-3} = \frac{-12}{x^2-9}$

Background

Topic: Solving Rational Equations with Unlike Denominators

This question tests your ability to solve rational equations by finding a common denominator and combining terms.

Key Terms and Formulas:

• Least Common Denominator (LCD): The smallest expression that is a common multiple of all denominators.

• Factoring: $x^2-9 = (x+3)(x-3)$

Step-by-Step Guidance

1. Factor the denominator on the right: $x^2-9 = (x+3)(x-3)$.

2. Identify the LCD for all terms: $(x+3)(x-3)$.

3. Multiply both sides of the equation by the LCD to clear denominators.

4. Simplify each term and solve the resulting equation for $x$.

5. Check for extraneous solutions by substituting back into the original denominators.

Try solving on your own before revealing the answer!