Algebra Notes D3 Unit 12

Skill Targets

  • Identify domain restrictions on rational equations.
  • Solve rational equations by multiplying by the LCD (least common denominator) first, then solving.
  • Solve rational inequalities.
  • Formulate rational equations that model real-world situations.

Quick Review - Rational Functions

  • Multiply the Rational Expressions:
    • Ensure expressions are defined (denominators cannot be zero).
  • Subtract the Rational Expressions:
    • Ensure expressions are defined.

Section I: Previous Concepts

  • Key strategy: Cross multiplying helps eliminate fractions but understanding why it works is vital.
  • LCD Method: Multiply both sides by the LCD to eliminate the fractions.

Section II: Solving Rational Equations

Equations with One Term on Each Side
  • Use cross multiplication as a strategy, but identify domain restrictions where denominators equal zero (these values cannot be solutions).
  • Example:
    • Given: ( rac{2x + 7}{x - 5} = 4 )
    • LCD is ( (x - 5) ) and domain restriction ( x \neq 5 ).
  • Procedure:
    • Multiply both sides by the LCD to eliminate fractions.
    • Example: ( (x - 5)(2x + 7) = 4(x - 5) )
    • Yielding a simplified equation: ( 30x - 60 = 15x + 120 )
    • Check solutions by substitution or graphically ( (1.5 = 1.5) ).

Section III: Solving Rational Equations with Multiple Terms

Steps to Solve
  1. Identify Domain Restrictions:
    • Set denominators with variables equal to zero.
  2. Determine the LCD:
    • List denominators and unique factors.
  3. Multiply:
    • Distribute the LCD to every term on both sides.
  4. Cancel and Rewrite:
    • Be cautious of negative signs during distribution.
  5. Solve for x:
    • Example Calculation: ( 3x + 3 - x + 3 = 6x - 18 ) simplifies to ( 4x = 24 ) leading to ( x = 6 )
  6. Check Solutions:
    • Verify algebraically and graphically.

Section IV: Modeling Real-World Situations

Example Problems
  1. Jet Travel:
    • Distance: 3950 miles round trip.
    • Total time: 16.5 hours.
    • Average speed without wind: 485 mi/h.
    • Question: Find average wind speed.
  2. Tank Cleaning:
    • Jason’s time alone: 6 hours.
    • Together with Lacy: 3.5 hours.
    • Question: Find Lacy’s time alone.
  3. Senior Class Statistics:
    • Total seniors: 400, increasing 10/year.
    • College attendees: 205, increasing 8/year.
    • Question: Current percentage and when it will reach 60%.

Conclusion

  • Always consider domain restrictions before solving rational equations.
  • Verify solutions to ensure they fall within the defined domain and are accurate.