Subtract.
$\frac{1}{x+7}-\frac{3}{x-7}$
A. $-\frac{2 x+28}{x^2-49}$
B. $\frac{4 x+14}{x^2-49}$
C. $-\frac{2}{x-7}$
D. $\frac{2}{x-7}$
Answer (1)
Find a common denominator for the two fractions: ( x + 7 ) ( x − 7 ) = x 2 − 49 .
Rewrite each fraction with the common denominator.
Subtract the two fractions: x 2 − 49 x − 7 − x 2 − 49 3 x + 21 = x 2 − 49 ( x − 7 ) − ( 3 x + 21 ) .
Simplify the result: x 2 − 49 − 2 x − 28 .
x 2 − 49 − 2 x − 28
Explanation
Problem Analysis We are asked to subtract two rational expressions. To do this, we need to find a common denominator and then combine the numerators.
Finding the Common Denominator The two rational expressions are x + 7 1 and x − 7 3 . The common denominator is ( x + 7 ) ( x − 7 ) , which is equal to x 2 − 49 .
Rewriting Fractions We rewrite each fraction with the common denominator: x + 7 1 = ( x + 7 ) ( x − 7 ) 1 ( x − 7 ) = x 2 − 49 x − 7 x − 7 3 = ( x − 7 ) ( x + 7 ) 3 ( x + 7 ) = x 2 − 49 3 x + 21
Subtracting the Fractions Now we subtract the two fractions: x + 7 1 − x − 7 3 = x 2 − 49 x − 7 − x 2 − 49 3 x + 21 = x 2 − 49 ( x − 7 ) − ( 3 x + 21 ) = x 2 − 49 x − 7 − 3 x − 21 = x 2 − 49 − 2 x − 28
Simplifying the Result We can factor out a -2 from the numerator: x 2 − 49 − 2 x − 28 = x 2 − 49 − 2 ( x + 14 )
Final Answer Comparing our result with the given options, we see that the correct answer is: x 2 − 49 − 2 x − 28
Examples
Rational expressions are used in various fields, such as physics, engineering, and economics, to model relationships between different quantities. For example, in physics, they can be used to describe the motion of objects or the behavior of electrical circuits. In economics, they can be used to model supply and demand curves or to analyze the effects of government policies. Understanding how to manipulate and simplify rational expressions is essential for solving problems in these fields.
