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Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions

Objectives

  • Adding and subtracting rational expressions with like denominators.
  • Adding and subtracting rational expressions with unlike denominators.
  • Always write the fraction in the simplest form.

Adding and Subtracting with Like Denominators

  • If the denominators of fractions are the same, add or subtract the numerators.

Example with Rational Numbers

  • Adding fractions
    • \frac{1}{5} + \frac{7}{5} = \frac{1+7}{5} = \frac{8}{5}
    • \frac{1}{5} is a proper fraction (numerator < denominator).
    • \frac{7}{5} is an improper fraction (numerator > denominator).
  • Subtracting fractions
    • \frac{1}{5} - \frac{7}{5} = \frac{1-7}{5} = \frac{-6}{5}
    • Integers: negative counting numbers, zero, and positive counting numbers.

Example with Algebraic Expressions

  • Example 1:
    • \frac{x}{4} + \frac{5-x}{4}
    • Denominators are the same.
    • Add the numerators: \frac{x + 5 - x}{4}
    • Simplify: \frac{5}{4}
    • Domain: All real numbers because the denominator (4) is different than zero, and there is no x in the denominator.
    • Interval notation: (-\infty, \infty)
  • Example 2:
    • \frac{x}{x^2 - 2x - 3} - \frac{3}{x^2 - 2x - 3}
    • Factor the denominator: x^2 - 2x - 3 = (x - 3)(x + 1)
      • Possible combinations for -3: -3 and 1, since -3 + 1 = -2
    • Rewrite the expression:
      • \frac{x}{(x - 3)(x + 1)} - \frac{3}{(x - 3)(x + 1)}
    • Subtract the numerators: \frac{x - 3}{(x - 3)(x + 1)}
    • Simplify: \frac{1}{x + 1}
    • Domain: Use the original factored form of the denominator, (x - 3)(x + 1) \neq 0
      • x \neq 3 and x \neq -1
      • Interval notation: (-\infty, -1) \cup (-1, 3) \cup (3, \infty)

Least Common Multiple (LCM)

Finding the LCM of Algebraic Expressions

  • Monomials: Find the LCM of the coefficients and the highest power of the variable.
    • Example: Find the LCM of 6x, 2x^2, 9x^3
      • Variable: x^3
      • Coefficients: 6, 2, and 9.
        • Multiples of 6: 6, 12, 18, 24, …
        • Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, …
        • Multiples of 9: 9, 18, 27, …
      • LCM of coefficients: 18
      • LCM of the algebraic expressions: 18x^3
  • Binomials: Factorize the expressions first.
    • Example: Find the LCM of x^2 - x and 2x - 2
      • Factorization:
        • x^2 - x = x(x - 1)
        • 2x - 2 = 2(x - 1)
      • Factors: 2, x, and x - 1.
      • LCM: 2 * x * (x-1) = 2x(x - 1)
  • Trinomials: Factorize the expressions first.
    • Example: Find the LCM of 3x^2 + 6x and x^2 + 4x + 4
      • Factorization:
        • 3x^2 + 6x = 3x(x + 2)
        • x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2
      • Factors: 3x and x + 2.
      • LCM: 3x(x + 2)^2

Adding and Subtracting with Unlike Denominators

  • Find the least common multiple (LCM) of the denominators.
  • Convert each fraction to an equivalent fraction with the LCM as the denominator.
  • Add or subtract the numerators.
  • Simplify the result.

Example 1

  • \frac{7}{6x} + \frac{5}{8x}
  • LCM of 6x and 8x:
    • LCM of 6 and 8: 24
    • LCM of x and x: x
    • LCM: 24x
  • Convert fractions:
    • \frac{7}{6x} * \frac{4}{4} = \frac{28}{24x}
    • \frac{5}{8x} * \frac{3}{3} = \frac{15}{24x}
  • Add the fractions: \frac{28}{24x} + \frac{15}{24x} = \frac{28 + 15}{24x} = \frac{43}{24x}

Example 2

  • \frac{2x}{x + 1} - \frac{x}{x + 2}
    • x cannot be -1 and -2 (given).
  • LCM of x + 1 and x + 2: (x + 1)(x + 2)
  • Convert fractions:
    • \frac{2x}{x + 1} * \frac{x + 2}{x + 2} = \frac{2x(x + 2)}{(x + 1)(x + 2)} = \frac{2x^2 + 4x}{(x + 1)(x + 2)}
    • \frac{x}{x + 2} * \frac{x + 1}{x + 1} = \frac{x(x + 1)}{(x + 1)(x + 2)} = \frac{x^2 + x}{(x + 1)(x + 2)}
  • Subtract the fractions:
    • \frac{2x^2 + 4x}{(x + 1)(x + 2)} - \frac{x^2 + x}{(x + 1)(x + 2)} = \frac{2x^2 + 4x - (x^2 + x)}{(x + 1)(x + 2)} = \frac{2x^2 + 4x - x^2 - x}{(x + 1)(x + 2)} = \frac{x^2 + 3x}{(x + 1)(x + 2)}
    • Domain: x cannot be -1 or -2.

Example 3

  • \frac{3}{x^2 - 1} + \frac{x}{x^2 + 2x + 1}
  • Factorize denominators:
    • x^2 - 1 = (x + 1)(x - 1)
    • x^2 + 2x + 1 = (x + 1)^2
  • LCM: (x + 1)^2(x - 1)
  • Convert fractions:
    • \frac{3}{(x + 1)(x - 1)} * \frac{x + 1}{x + 1} = \frac{3(x + 1)}{(x + 1)^2(x - 1)}
    • \frac{x}{(x + 1)^2} * \frac{x - 1}{x - 1} = \frac{x(x - 1)}{(x + 1)^2(x - 1)}
  • Add the fractions:
    • \frac{3(x + 1)}{(x + 1)^2(x - 1)} + \frac{x(x - 1)}{(x + 1)^2(x - 1)} = \frac{3x + 3 + x^2 - x}{(x + 1)^2(x - 1)} = \frac{x^2 + 2x + 3}{(x + 1)^2(x - 1)}
    • Domain: x cannot be -1 or 1.

Example 4

  • \frac{1}{x^2 - x} - \frac{3x}{4x^2 - 4x}
  • Simplify 4x^2 - 4x = 4x(x-1)
  • LCM between x^2 - x = x(x-1) and 4x(x-1) is 4x(x-1)^2
    • x must be different than 0 and 1.
  • Simplify the expression to find the original form: \frac{x^2 + 4x - 8}{4x(x - 1)^2}

Complex Fractions

  • A fraction over another fraction.
  • Represents the division of fractions.
  • Example: \frac{\frac{1}{2} + \frac{1}{x}}{\frac{x^2 - 4}{2x}}

Simplifying Complex Fractions

  • Simplify the numerator and denominator separately.
  • Rewrite the complex fraction as a division problem.
  • Multiply by the reciprocal of the denominator.
  • Simplify the result.
Example
  • \frac{\frac{1}{2} + \frac{1}{x}}{\frac{x^2 - 4}{2x}}
  • Simplify the numerator:
    • \frac{1}{2} + \frac{1}{x} = \frac{x}{2x} + \frac{2}{2x} = \frac{x + 2}{2x}
  • Rewrite the complex fraction as a division problem:
    • \frac{x + 2}{2x} \div \frac{x^2 - 4}{2x}
  • Multiply by the reciprocal of the denominator:
    • \frac{x + 2}{2x} * \frac{2x}{x^2 - 4} = \frac{x + 2}{2x} * \frac{2x}{(x + 2)(x - 2)}
  • Simplify the result:
    • \frac{x + 2}{2x} * \frac{2x}{(x + 2)(x - 2)} = \frac{1}{x - 2}
    • Restrictions: x cannot be 0, -2, or 2.
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