Select the correct answer.
Juan needs to rewrite this difference as one expression.
$\frac{3 x}{x^2-7 x+10}-\frac{2 x}{3 x-15}$
First he factored the denominators.
$\frac{3 x}{(x-2)(x-5)}-\frac{2 x}{3(x-5)}$
What step should Juan take next when subtracting these expressions?
A. Cancel the factor $x$ from the numerator and the denominator of both fractions.
B. Multiply the first fraction by $\frac{x-5}{x-5}$.
C. Subtract the numerators.
D. Multiply the second fraction by $\frac{x-2}{x-2}$
Answer (1)
To subtract the given rational expressions, Juan should first find a common denominator. The least common denominator is 3 ( x − 2 ) ( x − 5 ) . To obtain this LCD, he needs to multiply the second fraction by x − 2 x − 2 . Therefore, the correct answer is: $\boxed{D}
Explanation
Problem Analysis We are asked to determine the next step in subtracting two rational expressions. The given expression is x 2 − 7 x + 10 3 x − 3 x − 15 2 x . Juan has already factored the denominators as ( x − 2 ) ( x − 5 ) 3 x − 3 ( x − 5 ) 2 x . To subtract these expressions, we need to find a common denominator.
Finding the Least Common Denominator The denominators are ( x − 2 ) ( x − 5 ) and 3 ( x − 5 ) . The least common denominator (LCD) is 3 ( x − 2 ) ( x − 5 ) . To obtain the LCD in the first fraction, we need to multiply the denominator ( x − 2 ) ( x − 5 ) by 3 . Thus, we multiply the first fraction by 3 3 . To obtain the LCD in the second fraction, we need to multiply the denominator 3 ( x − 5 ) by ( x − 2 ) . Thus, we multiply the second fraction by x − 2 x − 2 .
Determining the Next Step The options are: A. Cancel the factor x from the numerator and the denominator of both fractions. B. Multiply the first fraction by x − 5 x − 5 .
C. Subtract the numerators. D. Multiply the second fraction by x − 2 x − 2 Option A is incorrect because we need a common denominator before we can subtract the fractions. Cancelling the factor x is not the correct next step. Option B is incorrect because we need to multiply the first fraction by 3 3 to obtain the LCD. Option C is incorrect because we need a common denominator before we can subtract the numerators. Option D is correct because we need to multiply the second fraction by x − 2 x − 2 to obtain the LCD.
Conclusion Therefore, the next step is to multiply the second fraction by x − 2 x − 2 .
Examples
When subtracting fractions in algebra, finding a common denominator is similar to ensuring you're comparing 'apples to apples.' For instance, if you're mixing paint colors, you need to express the amounts in the same units (e.g., ounces) before you can accurately determine the resulting color. Similarly, in recipe scaling, you need a common denominator to correctly adjust ingredient quantities. This principle ensures accurate and meaningful comparisons or combinations.
