Example Card: Adding Rational Expressions with Unlike Denominators
Add rational expressions with unlike denominators by finding LCD, converting fractions, and combining numerators. Work through Grade 9 algebraic fraction addition.
Key Concepts
Combining different algebraic fractions is like finding a common language. Let's see how with this key idea of adding rational expressions with unlike denominators.
Example Problem: Add the expressions $\frac{5x^2}{x^2 9} + \frac{x+2}{3x 9}$.
Step by Step: 1. First, factor each denominator to identify the building blocks. $$ \frac{5x^2}{(x 3)(x+3)} + \frac{x+2}{3(x 3)} $$ 2. The least common denominator (LCD) must contain every factor from both denominators. The LCD is $3(x 3)(x+3)$. 3. Now, write an equivalent fraction for each term using the LCD. For the first fraction, multiply the numerator and denominator by $3$: $$ \frac{5x^2}{(x 3)(x+3)} \cdot \frac{3}{3} = \frac{3(5x^2)}{3(x 3)(x+3)} $$ For the second fraction, multiply the numerator and denominator by $(x+3)$: $$ \frac{x+2}{3(x 3)} \cdot \frac{x+3}{x+3} = \frac{(x+2)(x+3)}{3(x 3)(x+3)} $$ 4. With common denominators, we can add the numerators. $$ \frac{3(5x^2) + (x+2)(x+3)}{3(x 3)(x+3)} $$ 5. Finally, expand and combine like terms in the numerator to simplify. $$ \frac{15x^2 + x^2 + 5x + 6}{3(x 3)(x+3)} = \frac{16x^2 + 5x + 6}{3(x 3)(x+3)} $$.
Common Questions
What is Adding Rational Expressions with Unlike Denominators in Grade 9 algebra?
It is a core concept in Grade 9 algebra that builds problem-solving skills and prepares students for advanced math coursework.
How do you apply adding rational expressions with unlike denominators to solve problems?
Identify the relevant formula or property, substitute known values carefully, apply each step in order, and verify the result makes sense.
What common errors occur with adding rational expressions with unlike denominators?
Misapplying the rule to wrong scenarios, sign mistakes, and forgetting to check answers in the original problem.