Which expression is equivalent to $\frac{2 n}{n+4}+\frac{7}{n-1}$ if no denominator equals zero?
A. $\frac{2 n^2+5 n+28}{(n+4)(n-1)}$
B. $\frac{2 n^2+5 n+4}{(n+4)(n-1)}$
C. $\frac{2 n^2+6 n+28}{(n+4)(n-1)}$
D. $\frac{2 n^2+6 n+4}{(n+4)(n-1)}$
Answer (1)
Find a common denominator: ( n + 4 ) ( n − 1 ) .
Rewrite the fractions with the common denominator and add them: ( n + 4 ) ( n − 1 ) 2 n ( n − 1 ) + 7 ( n + 4 ) .
Expand and simplify the numerator: ( n + 4 ) ( n − 1 ) 2 n 2 − 2 n + 7 n + 28 = ( n + 4 ) ( n − 1 ) 2 n 2 + 5 n + 28 .
The equivalent expression is ( n + 4 ) ( n − 1 ) 2 n 2 + 5 n + 28 .
Explanation
Problem Analysis We are given the expression n + 4 2 n + n − 1 7 and asked to find an equivalent expression. To do this, we need to combine the two fractions by finding a common denominator.
Finding Common Denominator The common denominator for the two fractions is ( n + 4 ) ( n − 1 ) . We rewrite each fraction with this common denominator: n + 4 2 n = ( n + 4 ) ( n − 1 ) 2 n ( n − 1 ) n − 1 7 = ( n + 4 ) ( n − 1 ) 7 ( n + 4 )
Adding Fractions Now we add the two fractions: ( n + 4 ) ( n − 1 ) 2 n ( n − 1 ) + ( n + 4 ) ( n − 1 ) 7 ( n + 4 ) = ( n + 4 ) ( n − 1 ) 2 n ( n − 1 ) + 7 ( n + 4 )
Expanding Numerator Next, we expand the numerator: ( n + 4 ) ( n − 1 ) 2 n 2 − 2 n + 7 n + 28
Simplifying Numerator Finally, we simplify the numerator by combining like terms: ( n + 4 ) ( n − 1 ) 2 n 2 + 5 n + 28
Final Answer Comparing our simplified expression with the given options, we see that it matches option A.\newlineTherefore, the equivalent expression is ( n + 4 ) ( n − 1 ) 2 n 2 + 5 n + 28 .
Examples
Understanding how to combine rational expressions is useful in many areas of math and science. For example, in physics, you might need to combine expressions when dealing with electrical circuits or fluid dynamics. In economics, you might use these skills when analyzing supply and demand curves. Knowing how to manipulate these expressions allows you to simplify complex models and make predictions about real-world phenomena.
