QQuestionMathematics
QuestionMathematics
Explain the steps involved in adding two rational expressions.
Find a common denominator: Identify the least common multiple (LCM) of the denominators of the two rational expressions.
Rewrite each fraction: Convert each rational expression to an equivalent expression with the common denominator.
Combine the numerators: Add the numerators of the equivalent fractions while keeping the common denominator.
Simplify the result: Simplify the resulting fraction, if possible, by factoring the numerator and denominator and reducing to lowest terms.
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Answer
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Step 1:I'll solve this problem by demonstrating how to add rational expressions step by step:
Step 2:: Understand the General Process
To add rational expressions, we must follow a systematic approach: - Find the least common multiple (LCM) of the denominators - Convert each fraction to an equivalent fraction with the common denominator - Add the numerators - Simplify the resulting fraction
Step 3:: Example Demonstration
Let's add $$\frac{2x}{x+3}$$ and $$\frac{3x}{x-1}
Step 4:: Find the Least Common Denominator (LCD)
- First, factor the denominators: $$(x+3)(x-1)
- The LCD is the product of these factors: \frac{2x}{x+ 3} + \frac{3x}{x- 1} = \frac{2x(x- 1)}{(x+ 3)(x- 1)} + \frac{3x(x+ 3)}{(x+ 3)(x- 1)}
Step 5:: Expand the Numerators
\frac{2x(x-1)}{(x+3)(x-1)} + \frac{3x(x+3)}{(x+3)(x-1)}
= \frac{2x^{2} - 2x}{(x+ 3)(x- 1)} + \frac{3x^{2} + 9x}{(x+ 3)(x- 1)}
Step 6:: Add the Numerators
= \frac{2x^{2} - 2x + 3x^{2} + 9x}{(x+3)(x-1)}
= \frac{5x^{2} + 7x}{(x+ 3)(x- 1)}
Step 7:: Simplify if Possible
In this case, the fraction cannot be simplified further.
Final Answer
Key Insights: - Always find the least common denominator - Multiply each numerator by the appropriate factor to create equivalent fractions - Add the numerators while keeping the common denominator - Simplify the result when possible
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