Lecture on Solving Rational Equations

Guidelines for Solving Rational Equations

1. Factor all denominators
2. Determine the LCD (Least Common Denominator) and multiply each side of the equation by the LCD.
3. Solve the resulting equation.
4. Check solutions:
   • Ensure the solution does not make any denominator zero.
   • Exclude such values from the solution.
5. Graphical Check:
   • Use a graphing calculator.
   • Input left side in Y1 and right side in Y2.
   • Check solutions by finding intersection points of the two graphs.

Example Problems

Example 1

• Equation: Not specified.
• LCD: 6T
• Steps:
  • Multiply each term by the LCD (6T/1).
  • Simplify products.
  • Combine like terms to solve.
  • Result: Simplified to -T = 6 which solves to T = -6.

Example 2

• Observation: Proportion problem.
• LCD: Product of two denominators.
• Steps:
  • Multiply both sides with the product of denominators.
  • Simplify and remove fractions.
  • Solve by FOIL method and combine terms.
  • Result: X = 2/11.
• Verification:
  • Check via graphing calculator.
  • Intersection point confirms the solution as 2/11.

Example 3

• Denominator Factorization: Required.
• LCD: X + 3 and X - 2.
• Steps:
  • Multiply each term by the LCD and simplify.
  • Solve resulting equation.
  • Result: X = -2.
  • Note: Check ensures no denominator equals zero for this value.

Example 4

• Denominator Factorization: Required.
• LCD: Y + 2 and Y - 1.
• Steps:
  • Multiply each rational expression by the LCD.
  • Simplify and solve.
  • Result: Y = 2.
  • Note: Check confirms Y = 2 does not make any denominator zero.

Conclusion

• Rational equations can be complex.
• Essential to factor denominators, determine the LCD, and verify solutions.
• Use graphical tools for verification when possible.
• The lecture provided step-by-step examples to illustrate the process.