Notes on Adding & Subtracting Rational Expressions

Unit 7.5 – Adding & Subtracting Rational Expressions

Adding and Subtracting Rational Expressions

  • Definition of Rational Expressions:

    • A rational expression is a fraction where the numerator and the denominator are polynomials.

Example 1: Simple Addition

  • Perform addition:

    • \frac{3}{9} + \frac{2}{9} = \frac{5}{9}

    • Subtraction:

    • \frac{7}{9} - \frac{2}{9} = \frac{5}{9}

Example 2: Adding Different Denominators

  • Adding rational expressions with different denominators:

    • Expression: \frac{3}{4} + \frac{3}{8}

    • Finding Common Denominator:

    • One of the denominators (4) divides evenly into the other (8):

      • 8 \div 4 = 2

    • Multiply the numerator and denominator of the first fraction by 2:

      • \frac{3}{4} \cdot \frac{2}{2} = \frac{6}{8}

    • Now add:

    • \frac{6}{8} + \frac{3}{8} = \frac{9}{8}

Example 3: Variables in Denominators

  • Expression: \frac{2x}{x + 6} - \frac{x}{x + 5}

  • To make the denominators equal:

    • Determine if one denominator divides evenly into the other:

    • Example: \frac{y^3}{y^2} = y

    • Here, multiply the numerator and denominator of the first fraction by y:

Finding the Least Common Denominator (LCD)

  • Identifying LCD:

    • The easiest approach is to multiply the denominators:

    • For example: \frac{3}{9} + \frac{1}{2}

    • Multiply by the other denominator:

    • \frac{3}{9} \cdot \frac{2}{2} = \frac{6}{18}

    • \frac{1}{2} \cdot \frac{9}{9} = \frac{9}{18}

    • Now add them together:

    • \frac{6}{18} + \frac{9}{18} = \frac{15}{18} = \frac{5}{6}

Simplifying Expressions After Operations

  • Performing Operations:

    • When one of the denominators does not divide evenly into the other, you can:

    • Multiply each fraction by the other denominator to retain fractions:

    • Example: Leave out common variable factors if applicable.

Skill Examples

Skill 1: Example Problem

  • Addition of rational expressions:

    • \frac{7}{(x-9)} + \frac{5}{(x-9)}

    • Combine to:(\frac{7x - 63 + 5}{(x-9)})

Skill 2: Example Problem with Rational Expressions

  • Given \frac{(x-8)(2)}{(x-3)(2)} and adding a variable expression.

Skill 3: Advanced Example Problem

  • Expression: \frac{7x}{(x-3)(x-8)} + \frac{(x-3)(x-8)}{(x-3)(x-8)}

    • Combine to:

    • The expression simplifies using the common denominator property.

Real-World Applications

Model Flight Time

  • Total Time Equation for Flight:

    • The time needed to fly from Jacksonville to Seattle and back is modeled by:

    • T = \frac{d}{a-j} + \frac{d}{a+j}

    • Where:

    • d: distance (in miles)

    • a: average airplane speed (in mph)

    • j: average jet stream speed (in mph)

  • Example Calculation:

    • Actual distance: 2438 miles with a speed of: a = 510, j = 115.

    • Substitute these values into the equation and simplify:

    • Find total time by using the developed model:

    • \text{Substituted: } T = \frac{2438}{510 - 115} + \frac{2438}{510 + 115}

Conclusion

  • By following these steps and applying the mentioned methods, rational expressions can be effectively added and subtracted, and the implications can be observed in practical applications such as flight modeling.