Question

Juan needs to rewrite this difference as one expression: (3x)/(x² - 7x + 10) - (2x)/(3x - 15). First, he factored the denominators. What is the simplified expression?

Options:
Option 1: (x - 2)/(x - 5)
Option 2: (3x - 5)/(x - 2)
Option 3: (2x - 3)/(x - 5)
Option 4: (3x - 2)/(x - 5)

Answer 1

To rewrite the difference as one expression, we need to find a common denominator for the two fractions. The simplified expression is (13x - 2x^2)/(3(x - 2)(x - 5)).

Step-by-step explanation:

To rewrite the difference as one expression, we need to find a common denominator for the two fractions. The denominators are already factored, which is a good start. The first denominator is x2 - 7x + 10, which can be factored as (x - 2)(x - 5). The second denominator is 3x - 15, which can be factored as 3(x - 5). Now, we can rewrite the expression with a common denominator of (x - 2)(x - 5):

(3x)/[(x - 2)(x - 5)] - (2x)/[3(x - 5)]

Next, we multiply the first fraction by 3/3 and the second fraction by (x - 2)/(x - 2) to obtain a common denominator:

(9x)/[3(x - 2)(x - 5)] - (2x(x - 2))/[3(x - 2)(x - 5)]

Simplifying, we get:

(9x - 2x(x - 2))/[3(x - 2)(x - 5)]

Expanding the expression in the numerator, we get:

(9x - 2x^2 + 4x)/[3(x - 2)(x - 5)]

Combining like terms, we have:

(13x - 2x^2)/[3(x - 2)(x - 5)]

So, the simplified expression is (13x - 2x^2)/[3(x - 2)(x - 5)].