Chapter 3 Review Study Guide

Problem-Solving Directions

  • To effectively solve these problems, always demonstrate all your work step-by-step. This helps identify errors and reinforces understanding.

  • Consider rewriting equations into simpler forms or using substitution methods when appropriate.

  • Wherever possible, verify your solutions by plugging them back into the original equation or system. This ensures accuracy.

  • A special note on prime notation (e.g., x'): In standard algebra, prime notation usually indicates differentiation (calculus). However, given the context of other problems in this review, it is highly probable that x' is a typo for x or denotes a distinct variable within an equation that should be solved for in terms of another. For single equations (like E and F below), we will assume x' is meant to be x or a similar variable to allow solving for a single value. If it's a differential equation, that would be a different topic.

Equations to Solve

How to approach:

  1. Isolate the variable: The goal is to get the variable (e.g., x) by itself on one side of the equation.

  2. Use inverse operations: To move terms, apply the opposite operation (e.g., if a number is added, subtract it; if multiplied, divide).

  3. Combine like terms: Simplify each side of the equation before isolating the variable.

  4. Distribute: If parentheses are present, distribute any coefficients.

  • E. 40x' - 10x - 50 = 0 (Assuming x' is a typo for x): 40x - 10x - 50 = 0 \implies 30x - 50 = 0 \implies 30x = 50 \implies x = 50/30 = 5/3

  • F. 5x' - 45x - 50 = 10 (Assuming x' is a typo for x): 5x - 45x - 50 = 10 \implies -40x - 50 = 10 \implies -40x = 60 \implies x = 60 / (-40) = -3/2

  • G. 300x - 200y = 600

  • H. -15x - 5y = -30

  • Additional Equations:

    • 100x + 400y = 1600

    • 5x - y = -10

  • Given values: x = 2, y = 0

Additional practice problems:

  1. Solve for z: 7(2z - 5) + 3z = 10z + 5

  2. Solve for w: rac{3w}{4} - 2 = rac{w}{2} + 1

  3. Solve for m: 0.25(m + 8) = 0.5m - 1

Simplification Problems

How to approach:

  1. Factor: Look for common factors in the numerator and denominator, or factor polynomials (e.g., trinomials, difference of squares).

  2. Cancel common terms: After factoring, cancel out identical factors from the numerator and denominator.

  3. Identify excluded values: Any value of the variable that makes the original denominator zero is an excluded value from the domain. List these before simplifying.

  • General Instructions: Simplify the given expressions fully. List any excluded values from the domain.

  • Original Problems (re-evaluated and valid ones emphasized):

    1. rac{9}{6x^7} (Simplify coefficients and power; excluded value: x \neq 0)

    2. rac{45x - 2400}{4} (Factor out common terms in numerator; no excluded values)

    3. rac{x^3 + 5x - 24}{x + 5x - 6} (Simplify denominator to 6x - 6. Factor numerator if possible, then perform division; excluded value: x \neq 1)

    4. x^2 + 5x + 6 (Factor as (x+2)(x+3); no excluded values)

    5. rac{7x^2 - 48}{x^3 - 18} (Check if numerator or denominator can be factored to find common terms; excluded value: x \neq \sqrt[3]{18})

    6. rac{55}{13} (Already simplified; no excluded values)

    7. rac{x^4 - 9}{x - 2} (Factor numerator as difference of squares; perform polynomial division or acknowledge no factors cancel; excluded value: x \neq 2)

    8. x^2

    9. x^4 + 7x - 10

Additional practice problems:

  • Simplify fully and list excluded values from the domain:

    1. rac{x^2 - 4x - 12}{x^2 - 3x - 18}

    2. rac{3x^2 - 12}{x^2 - 4x + 4}

    3. rac{x^3 - 8}{x^2 + 2x + 4}

Additional Algebraic Expressions (Expansion & Equivalence)

How to approach:

  1. Expand: Use the distributive property (e.g., FOIL method for two binomials) to multiply terms.

  2. Combine like terms: After expanding, group terms with the same variable and exponent.

  3. Determine Equivalence: To check if two expressions are equivalent, simplify both sides fully. If they simplify to the exact same expression, they are equivalent. Alternatively, substitute a few different numbers for the variable(s) into both expressions; if they always yield the same result, they are likely equivalent.

  • Original Problems (re-evaluated and valid ones emphasized):

    • 12. Simplify: 5(x - 8) + 14 (Assuming (x-8)' means (x-8), as prime notation for simplification isn't standard here)

    • 5x - 40 + 14 = 5x - 26

    • 15. Justification of Equivalence: Clarify logic that determines equivalency between expressions. (Already covered above in