Solving Fractional Equations Overview

Overview

• Topic: Solving fractional equations (equations with unknown(s) in denominators).
• Two main methods depending on equation form: cross-multiplication for single fractions each side, and combining fractions (or multiply by LCM) when multiple fractions appear.
• Includes examples converting fractional equations to linear or quadratic equations.

Cross-Product Property

• Statement: If A/B = C/D then A·D = B·C.
• Use only when there is a single fraction on each side of the equation.
• Converts fractional equation to non-fractional equation for easier solving.

Single Fractions On Each Side

• Strategy: Rewrite both sides as single simple fractions (put denominator 1 for integer terms), then apply cross-product.

• Steps:

  • Ensure each side is a single fraction.
  • Apply cross-multiplication AD = BC.
  • Solve the resulting linear equation by isolating the variable.

• Examples:

  • Example a) 3/ x = 9/20
    • Cross-multiply: 3·20 = 9·x → 60 = 9x → x = 60/9 = 20/3.
  • Example b) x/2(x+2?) [interpreted from transcript as (x/2) = (x+2)/? Actually presented: x/2 = (x+2)·3/5?]
    • Process demonstrated: cross-multiply, expand, isolate variable, divide.
    • Final shown: x = 8.
  • Example c) 3/(x-3)? = 4/(x-5)? [transcript shows 3/(x-3) = 4/(x-5)]
    • Cross-multiply, expand, isolate, divide.
    • Final shown: x = 3.

• Note: If one side is not a fraction, write it as something/1 before cross-multiplying.

  • Example: 3/x = 15. Rewritten 3/x = 15/1, cross-multiply to solve.

Multiple Fractions On Either Side

• Strategy A (combine fractions on a side):

  • Use LCM of denominators to combine multiple fractions into a single fraction on that side.
  • Then rewrite entire equation as single fraction vs single fraction and apply cross-multiplication.
  • Solve resulting linear or quadratic equation as required.

• Strategy B (multiply by LCM of all denominators):

  • Multiply every term of the equation by the LCM of all denominators to clear fractions at once.
  • Solve the resulting polynomial equation.
  • Ensure to consider domain restrictions (values making denominators zero) — avoid extraneous roots.

• Examples:

  • Example d) 3/4 + 1/(8x) = 0
    • LCM of denominators on left: 8x.
    • Rewrite: (3·2x)/(8x) + (1)/(8x) = 0 → (6x + 1)/(8x) = 0.
    • Cross-multiply with 0 as 0/1: (6x + 1)/ (8x) = 0/1 → 6x + 1 = 0 → x = -1/6? (transcript shows x = 1/6; sign in original steps ambiguous)
    • Solve linear equation after clearing fractions.
  • Example e) x/6 + 2/(3x) = 2/3
    • LCM of denominators on LHS: 6x.
    • Rewrite each term over 6x: (x^2)/(6x) + (4)/(6x) = 2/3 → combine LHS: (x^2 + 4)/(6x) = 2/3.
    • Cross-multiply: 3(x^2 + 4) = 12x → 3x^2 + 12 = 12x → standard quadratic: 3x^2 - 12x + 12 = 0.
    • Factor (if possible): 3(x^2 - 4x + 4) = 0 → 3(x - 2)^2 = 0 → x = 2 (double root).
    • Check that x does not make any denominator zero.

• Important: After multiplying by denominators or using cross-multiplication, always check for extraneous solutions that make original denominators zero.

Key Terms And Definitions

• Fractional Equation: Equation where the unknown appears in the denominator of one or more terms.
• Cross-Product Property: For A/B = C/D, AD = BC.
• LCM (Least Common Multiple): Smallest common multiple of denominators used to combine or clear fractions.
• Extraneous Root: Solution obtained after manipulation that does not satisfy the original equation (often because it makes a denominator zero).

Worked Example Table

| Problem | Method | Main Steps | Result | | 3/x = 9/20 | Cross-multiply | 3·20 = 9·x → 60 = 9x → x = 60/9 | x = 20/3 | | 3/x = 15 | Put 15 as fraction 15/1, cross-multiply | 3·1 = 15·x → 3 = 15x → x = 3/15 | x = 1/5 | | 3/4 + 1/(8x) = 0 | Combine LHS fractions then cross-multiply | LCM 8x → (6x + 1)/(8x) = 0 → 6x + 1 = 0 | x = -1/6 (check domain) | | x/6 + 2/(3x) = 2/3 | Combine to one fraction, cross-multiply → quadratic | LHS → (x^2 + 4)/(6x) = 2/3 → 3(x^2 + 4) = 12x → 3(x - 2)^2 = 0 | x = 2 |

Tips And Common Pitfalls

• Always rewrite non-fraction terms as something/1 before cross-multiplying.
• When combining fractions, compute the correct LCM and convert each fraction carefully.
• After clearing denominators, simplify and solve the resulting linear or quadratic equation.
• Check candidate solutions in the original equation to avoid accepting values that make denominators zero.
• For quadratics resulting from clearing denominators, factor or use quadratic formula as needed; watch for repeated roots.

Exit Problem (Practice)

• Solve: (2)/(x+1) + 3 = 1/2
• Suggested approach:
  • Rewrite 3 as 3/1, convert all terms to fractions with common denominator (or multiply both sides by LCM 2(x+1)).
  • Clear denominators, solve the resulting linear equation, then check domain (x ≠ -1).