MAT1033 - Gubitti REVIEW for Test #3 Study Notes

Chapter Overview

• Focus on rational expressions, simplifying, adding/subtracting, multiplying, dividing, solving equations, and understanding domain restrictions.

Section 1: Excluding Values from Expressions

1. Determine Excluded Values

   • Expression: $f(x) = rac{x}{9x - 2}$

   • Excluded values determined by setting the denominator to zero:

     • $9x - 2 = 0$ gives $x = rac{2}{9}$

   • Answer: A) $rac{-2}{9}, 0$

Section 2: Finding the Domain of Rational Functions

2. Domain of the Function

   • Expression: $f(x) = rac{x}{9x - 7}$

   • Denominator cannot be zero:

     • $9x - 7 <br>eq 0$ leads to $x <br>eq rac{7}{9}$

   • Constraints: $x ext{ is a real number and } x <br>eq rac{7}{9}$

   • Answer: B) $x | x ext{ is a real number and } x <br>eq rac{7}{9}, x <br>eq 0$

Section 3: Multiplication of Expressions

3. Multiply Expressions

   • Expression: $2p - 2p imes 3p^2 / 7p - 7$

   • Calculation Procedure:

     • Distributive property applied to simplify the multiplication.

     • Answer: B) $6p^3 - 6p^2 / 7p^2 - 7p$

4. Another Multiplication Example

   • Expression: $rac{6x - 9x^2}{9x^2 + 30x + 25} imes (9x^2 + 21x + 10)$

   • Answer: A) $3x(3x + 5)$

Section 4: Division of Expressions

5. Divide Expressions

   • Expression: $rac{x - y}{xy} ext{ divided by } (x^2 - y^2) / 8x^2y$

   • Simplification Procedure:

     • Use of the quotient rule for division and factorization.

   • Answer: B) $8x / (x + y)$

Section 5: Addition and Subtraction

6. Addition of Rational Expressions

   • Expression: $rac{4b + 20}{5a + 15} ÷ (ab - 3b + 5a - 15)$

   • Apply factorization and simplification techniques.

   • Answer: A) $rac{4(a - 3)}{5(a + 3)}$

Section 6: Simplifying Complex Fractions

7. Simplify the Complex Fraction

   • Expression: $rac{4x + 7x^2}{16x^2 - 49x}$

   • Utilize common factors and simplified forms.

   • Answer: B) $rac{1}{4x - 7}$

Section 7: Solving Algebraic Equations

8. Work-Rate Problem

   • Scenario: Painter vs. assistant.

   • Equation Formation:

     • Rate of painter: $rac{1}{4}$ house/hour

     • Rate of assistant: $rac{1}{6}$ house/hour

     • Combined Rate: $rac{1}{4} + rac{1}{6}$

     • Resulting in: $x = rac{12}{5} = 2.4 ext{ hours}$

   • Answer: B) $rac{5}{12} ext{ hours}$

9. Algebraic Equation

   • Equation: $rac{5}{x - 9} = rac{10}{x + 6}$

   • Solve by cross-multiplying.

   • Answer: A) $24$

Section 8: Advanced Algebraic Techniques

10. Equation: $rac{1}{x} + 4 - 7x - 4 = 4x^2 - 16$

    • Bring all terms to a common form, consolidate, and solve for $x$.

    • Answer: B) $-6$

11. Final Equation: $1 + rac{1}{x} = rac{12}{x^2}$

    • Rearranging and clearing denominators to form a polynomial, then factor.

    • Answer: D) $4, 3$

Conclusion: Study Tips

• Review the rules of operations in algebra: addition, subtraction, multiplication, and division.

• Practice solving various forms of equations to become familiar with manipulating expressions.

• Focus on understanding and applying factorization for simplification of rational functions and fractions.

• Solve practice problems similar to those in the review to build confidence for the test.