Simplify: \(\frac{x+2}{x+3} - \frac{x}{x-1}\)
Understand the Problem
The question asks to simplify the expression (\frac{x+2}{x+3} - \frac{x}{x-1}). This involves finding a common denominator, combining the fractions, and simplifying the resulting expression.
Answer
$\frac{-2x - 2}{x^2 + 2x - 3}$
Answer for screen readers
$\frac{-2x - 2}{x^2 + 2x - 3}$
Steps to Solve
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Find a common denominator The common denominator for the two fractions is the product of their individual denominators, which is $(x+3)(x-1)$.
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Rewrite each fraction with the common denominator Multiply the first fraction by $\frac{x-1}{x-1}$ and the second fraction by $\frac{x+3}{x+3}$: $$ \frac{x+2}{x+3} \cdot \frac{x-1}{x-1} - \frac{x}{x-1} \cdot \frac{x+3}{x+3} $$
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Expand the numerators Expand the products in the numerators: $$ \frac{(x+2)(x-1)}{(x+3)(x-1)} - \frac{x(x+3)}{(x-1)(x+3)} = \frac{x^2 + x - 2}{x^2 + 2x - 3} - \frac{x^2 + 3x}{x^2 + 2x - 3} $$
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Combine the fractions Since the fractions now have a common denominator, subtract the numerators: $$ \frac{(x^2 + x - 2) - (x^2 + 3x)}{x^2 + 2x - 3} $$
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Simplify the numerator Combine like terms in the numerator: $$ \frac{x^2 + x - 2 - x^2 - 3x}{x^2 + 2x - 3} = \frac{-2x - 2}{x^2 + 2x - 3} $$
$\frac{-2x - 2}{x^2 + 2x - 3}$
More Information
The simplified form of the given expression is $\frac{-2x - 2}{x^2 + 2x - 3}$.
Tips
- Incorrectly expanding the products: A common mistake is to make errors when expanding $(x+2)(x-1)$ or $x(x+3)$. Double-check the expansion to avoid such errors.
- Forgetting to distribute the negative sign: When subtracting the second fraction, remember to distribute the negative sign to all terms in the numerator. For example, $-(x^2 + 3x) = -x^2 - 3x$.
- Incorrectly combining like terms: Ensure correct combination of like terms in the numerator.
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