Add or subtract.
$\frac{-x}{-3}-\frac{2 x-5}{3-x}$
A. $-\frac{x+2}{x-3}$
B. $\frac{x+2}{x-3}$
C. $\frac{x+12}{x-3}$
D. $-\frac{x+12}{x-3}$
Answer (1)
Simplify the first term: − 3 − x = 3 x .
Rewrite the expression: 3 x − 3 − x 2 x − 5 .
Find a common denominator: 3 ( 3 − x ) .
Combine the fractions and simplify: 3 ( x − 3 ) x 2 + 3 x − 15 .
Explanation
Problem Analysis We are asked to simplify the expression − 3 − x − 3 − x 2 x − 5 and choose the correct answer from the provided options.
Simplifying the First Term First, simplify the first term: − 3 − x = 3 x .
Rewriting the Expression Rewrite the expression: 3 x − 3 − x 2 x − 5 .
Finding a Common Denominator Find a common denominator: The common denominator is 3 ( 3 − x ) .
Rewriting with Common Denominator Rewrite each fraction with the common denominator: 3 ( 3 − x ) x ( 3 − x ) − 3 ( 3 − x ) 3 ( 2 x − 5 ) .
Combining the Fractions Combine the fractions: 3 ( 3 − x ) x ( 3 − x ) − 3 ( 2 x − 5 ) .
Expanding the Numerator Expand the numerator: 3 ( 3 − x ) 3 x − x 2 − 6 x + 15 .
Simplifying the Numerator Simplify the numerator: 3 ( 3 − x ) − x 2 − 3 x + 15 .
Multiplying by -1 Multiply the numerator and denominator by -1: 3 ( x − 3 ) x 2 + 3 x − 15 .
Final Simplification and Comparison The simplified expression is 3 ( x − 3 ) x 2 + 3 x − 15 . None of the given options match this expression. Let's re-examine the steps to see if there was an error. The original expression is − 3 − x − 3 − x 2 x − 5 = 3 x − 3 − x 2 x − 5 . The common denominator is 3 ( 3 − x ) . So we have 3 ( 3 − x ) x ( 3 − x ) − 3 ( 2 x − 5 ) = 3 ( 3 − x ) 3 x − x 2 − 6 x + 15 = 3 ( 3 − x ) − x 2 − 3 x + 15 . Multiplying top and bottom by -1 gives 3 ( x − 3 ) x 2 + 3 x − 15 . This does not match any of the options. However, if the original problem was 3 − x − 3 − x 2 x − 5 , then we would have 3 ( 3 − x ) − x ( 3 − x ) − 3 ( 2 x − 5 ) = 3 ( 3 − x ) − 3 x + x 2 − 6 x + 15 = 3 ( 3 − x ) x 2 − 9 x + 15 . This still doesn't match. Let's assume the problem was 3 x − x − 3 2 x − 5 . Then we have 3 ( x − 3 ) x ( x − 3 ) − 3 ( 2 x − 5 ) = 3 ( x − 3 ) x 2 − 3 x − 6 x + 15 = 3 ( x − 3 ) x 2 − 9 x + 15 . Still no match. Let's try 3 x − 3 − x 2 x + 5 = 3 ( 3 − x ) x ( 3 − x ) − 3 ( 2 x + 5 ) = 3 ( 3 − x ) 3 x − x 2 − 6 x − 15 = 3 ( 3 − x ) − x 2 − 3 x − 15 = 3 ( 3 − x ) − ( x 2 + 3 x + 15 ) = 3 ( x − 3 ) x 2 + 3 x + 15 . Still no match.
Examples
Simplifying rational expressions is a fundamental skill in algebra and calculus. For instance, when analyzing the behavior of complex systems in physics or engineering, you often encounter rational functions. Simplifying these functions allows you to more easily identify key characteristics, such as asymptotes or points of discontinuity, which can be crucial for understanding the system's stability and response to different conditions. This skill is also essential in economics for modeling cost and revenue functions to optimize business strategies.
