In Exercises 33–68, add or subtract as indicated. 3x/(x−3) − (x+4)/(x+2)

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Identify the two rational expressions to be subtracted: \(\frac{3x}{x-3} - \frac{x+4}{x+2}\).

Find the least common denominator (LCD) of the two fractions, which is the product of the distinct denominators: \((x-3)(x+2)\).

Rewrite each fraction with the LCD as the new denominator by multiplying numerator and denominator accordingly: \(\frac{3x(x+2)}{(x-3)(x+2)} - \frac{(x+4)(x-3)}{(x+2)(x-3)}\).

Combine the two fractions into a single fraction by subtracting the numerators over the common denominator: \(\frac{3x(x+2) - (x+4)(x-3)}{(x-3)(x+2)}\).

Expand the numerators, simplify by combining like terms, and write the final expression as a single simplified rational expression.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including simplifying and performing operations like addition or subtraction, is essential for solving problems involving rational expressions.

Finding a Common Denominator

To add or subtract rational expressions, you must find a common denominator, typically the least common denominator (LCD). This involves factoring denominators and determining the smallest expression that both denominators divide into, allowing the expressions to be combined.

Combining and Simplifying Rational Expressions

After rewriting expressions with a common denominator, combine the numerators by addition or subtraction. Then, simplify the resulting rational expression by factoring and reducing common factors to express the answer in simplest form.