Solving Rational Equations

Example 1: Solving ( \frac{5}{8} - \frac{3}{5} = \frac{x}{10} )

• Objective: Find the value of ( x ).
• Step 1: Eliminate fractions using the least common multiple (LCM) of the denominators (8, 5, 10).
  • LCM is 40.
  • Multiply each term by 40:
    • ( \frac{5}{8} \times 40 = 25 )
    • ( \frac{3}{5} \times 40 = 24 )
    • ( \frac{x}{10} \times 40 = 4x )
• Step 2: Set equation and solve:
  • ( 25 - 24 = 4x )
  • ( x = \frac{1}{4} )

Example 2: Solving ( \frac{x+8}{x} = 6 )

• Objective: Solve for ( x ).
• Step 1: Multiply each side by ( x ) to clear the fraction:
  • ( x^2 + 8 = 6x )
• Step 2: Rearrange and factor:
  • ( x^2 - 6x + 8 = 0 )
  • Factors: ( (x-4)(x-2) )
  • Solutions: ( x = 4, 2 )

Example 3: Solving ( \frac{x+3}{x-3} = \frac{12}{3} )

• Objective: Solve for ( x ) by cross-multiplying.
• Cross-multiply:
  • ( 12(x-3) = 3(x+3) )
  • Expand: ( 12x - 36 = 3x + 9 )
• Step 2: Simplify and solve:
  • ( 9x = 45 )
  • ( x = 5 )

Example 4: Solving ( \frac{9}{x} = \frac{x}{4} )

• Objective: Solve for ( x ) using cross-multiplication.
• Cross-multiply:
  • ( x^2 = 36 )
• Solution: ( x = \pm 6 )

Example 5: Solving ( \frac{4}{x-3} = \frac{9}{x+2} )

• Objective: Solve for ( x ) using cross-multiplication.
• Steps:
  • Cross-multiply: ( 4(x+2) = 9(x-3) )
  • Expand and simplify:
    • ( 4x + 8 = 9x - 27 )
    • Rearrange: ( 5x = 35 )
  • Solution: ( x = 7 )

Example 6: Solving ( \frac{x+2}{3} + 4 = \frac{x+9}{2} )

• Objective: Solve for ( x ) by eliminating fractions.
• Step 1: Use LCM of denominators (2 and 3), which is 6.
• Step 2: Clear fractions:
  • ( 2(x+2) + 24 = 3(x+9) )
  • Expand and solve:
    • ( 2x + 4 + 24 = 3x + 27 )
    • Simplify: ( x = 1 )

Example 7: Solving ( \frac{4}{x} + \frac{8}{x+2} = 4 )

• Objective: Solve for ( x ) using a common denominator.
• Step 1: Common denominator is ( x(x+2) ).
• Step 2: Rewrite and simplify:
  • ( 4(x+2) + 8x = 4x(x+2) )
  • Simplify and rearrange:
    • ( 4x + 8 + 8x = 4x^2 + 8x )
    • ( 4x^2 - 4x - 8 = 0 )
• Step 3: Factor and find solutions:
  • Factor: ( (x-2)(x+1) )
  • Solutions: ( x = 2, -1 )

Example 8: Solving ( \frac{x}{x+5} - \frac{5}{x-5} = \frac{14}{x^2-25} )

• Objective: Solve for ( x ) by clearing denominators.
• Step 1: Factor ( x^2-25 ) as ((x+5)(x-5)).
• Step 2: Clear fractions using common denominator ((x+5)(x-5)).
• Step 3: Expand and simplify:
  • ( x(x-5) - 5(x+5) = 14 )
  • Simplify: ( x^2 - 5x - 5x - 25 = 14 )
• Step 4: Rearrange and factor:
  • ( x^2 - 10x - 39 = 0 )
  • Factor: ( (x-13)(x+3) )
  • Solutions: ( x = 13, -3 )