Solving Rational Equations

Introduction

• Focus on solving rational equations by clearing fractions.
• Use least common multiple (LCM) to simplify equations.

Example Problems

Example 1: Simple Equation

• Equation: ( \frac{5}{8} - \frac{3}{5} = \frac{x}{10} )
• LCM of 8, 5, and 10: 40
• Multiply each term by 40:
  • ( \frac{5}{8} \times 40 = 25 )
  • ( \frac{3}{5} \times 40 = 24 )
  • ( \frac{x}{10} \times 40 = 4x )
• Equation becomes: 25 - 24 = 4x
• Solve: ( x = \frac{1}{4} )

Example 2: Quadratic Equation

• Equation: ( \frac{x+8}{x} = 6 )
• Multiply both sides by x:
  • ( x \times x = x^2 )
  • ( \frac{8}{x} \times x = 8 )
  • ( 6 \times x = 6x )
• Rearrange: ( x^2 - 6x + 8 = 0 )
• Factor: ((x-4)(x-2) = 0)
• Solutions: ( x = 4, 2 )

Example 3: Cross Multiplication

• Equation: ( \frac{x+3}{x-3} = \frac{12}{3} )
• Cross multiply:
  • ( 12(x-3) = 12x - 36 )
  • ( 3(x+3) = 3x + 9 )
• Simplify: 9x = 45
• Solve: ( x = 5 )

Example 4: Square Roots

• Equation: ( \frac{9}{x} = \frac{x}{4} )
• Cross multiply: ( x^2 = 36 )
• Solve for x: ( x = \pm 6 )

Example 5: Cross Multiplication

• Equation: ( \frac{4}{x-3} = \frac{9}{x+2} )
• Cross multiply:
  • ( 4(x+2) = 4x + 8 )
  • ( 9(x-3) = 9x - 27 )
• Solve: ( x = 7 )

Example 6: Least Common Multiple

• Equation: ( \frac{x+2}{3} + 4 = \frac{x+9}{2} )
• LCM of 2 and 3 is 6
• Multiply everything by 6 and simplify:
  • Result: ( 2(x+2) + 24 = 3(x+9) )
  • Simplify to: ( x = 1 )

Example 7: Common Denominator

• Equation: ( \frac{4}{x} + \frac{8}{x+2} = 4 )
• Common denominator: ( x(x+2) )
• Multiply through and simplify:
  • Rearrange and factor: ( 4x^2 - 4x - 8 = 0 )
  • Solutions: ( x = 2, -1 )

Example 8: Difference of Squares

• Equation: ( \frac{x}{x+5} - \frac{5}{x-5} = \frac{14}{x^2-25} )
• Factor Denominator: ( x^2-25 = (x+5)(x-5) )
• Multiply through by common denominator and simplify:
  • Solve: ( x^2 - 10x - 39 = 0 )
  • Solutions: ( x = 13, -3 )

Summary

• Use LCM to eliminate fractions.
• Cross multiply when dealing with equations of two fractions.
• Factor when necessary to solve quadratic equations.
• Always check for possible restrictions in the domain (e.g., division by zero).