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Find the sum: [tex]$\frac{3 x}{x^2-x-12}+\frac{x+1}{2 x+6}$[/tex]
A) [tex]$\frac{(x+2)(x-2)}{(x-4)(x+3)}$[/tex]
B) [tex]$\frac{4 x+1}{2(x-4)(x+3)}$[/tex]
C) [tex]$\frac{(x+4)(x-1)}{2(x-4)(x+3)}$[/tex]
Answer (2)
Factor the denominators of the given expression.
Find the least common denominator (LCD).
Rewrite each fraction with the LCD and add the fractions.
Simplify the numerator and factor to obtain the final answer: 2 ( x − 4 ) ( x + 3 ) ( x + 4 ) ( x − 1 )
Explanation
Problem Analysis We are asked to find the sum of two rational expressions: x 2 − x − 12 3 x + 2 x + 6 x + 1 . To do this, we need to find a common denominator and combine the numerators.
Factoring Denominators First, let's factor the denominators of both expressions. The first denominator is x 2 − x − 12 . We are looking for two numbers that multiply to -12 and add to -1. These numbers are -4 and 3. So, x 2 − x − 12 = ( x − 4 ) ( x + 3 ) . The second denominator is 2 x + 6 . We can factor out a 2 to get 2 x + 6 = 2 ( x + 3 ) .
Finding the LCD Now we can rewrite the original expression as ( x − 4 ) ( x + 3 ) 3 x + 2 ( x + 3 ) x + 1 . The least common denominator (LCD) of these two fractions is 2 ( x − 4 ) ( x + 3 ) .
Rewriting with LCD Next, we rewrite each fraction with the LCD. For the first fraction, we multiply the numerator and denominator by 2: ( x − 4 ) ( x + 3 ) 3 x ⋅ 2 2 = 2 ( x − 4 ) ( x + 3 ) 6 x . For the second fraction, we multiply the numerator and denominator by ( x − 4 ) : 2 ( x + 3 ) x + 1 ⋅ x − 4 x − 4 = 2 ( x − 4 ) ( x + 3 ) ( x + 1 ) ( x − 4 ) .
Adding Fractions Now we can add the two fractions: 2 ( x − 4 ) ( x + 3 ) 6 x + 2 ( x − 4 ) ( x + 3 ) ( x + 1 ) ( x − 4 ) = 2 ( x − 4 ) ( x + 3 ) 6 x + ( x + 1 ) ( x − 4 ) .
Expanding the Numerator Next, we expand the numerator: 6 x + ( x + 1 ) ( x − 4 ) = 6 x + ( x 2 − 4 x + x − 4 ) = 6 x + x 2 − 3 x − 4 = x 2 + 3 x − 4 . So the expression becomes 2 ( x − 4 ) ( x + 3 ) x 2 + 3 x − 4 .
Factoring the Numerator Now we factor the numerator: x 2 + 3 x − 4 . We are looking for two numbers that multiply to -4 and add to 3. These numbers are 4 and -1. So, x 2 + 3 x − 4 = ( x + 4 ) ( x − 1 ) . Thus, the expression becomes 2 ( x − 4 ) ( x + 3 ) ( x + 4 ) ( x − 1 ) .
Final Answer Therefore, the simplified sum is 2 ( x − 4 ) ( x + 3 ) ( x + 4 ) ( x − 1 ) . Comparing this to the given options, we see that it matches option C.
Examples
Rational expressions are useful in many real-world applications, such as calculating the cost of producing a certain number of items. For example, if the cost of producing x items is given by the expression 2 ( x − 4 ) ( x + 3 ) x 2 + 3 x − 4 , then simplifying this expression can help us understand how the cost changes as the number of items produced increases. Another application is in physics, where rational expressions can be used to model the motion of objects.
To add the two rational expressions, we factored the denominators and found the least common denominator (LCD). After rewriting each fraction with the LCD and combining them, we simplified and factored the result. The final answer, matching option C, is 2 ( x − 4 ) ( x + 3 ) ( x + 4 ) ( x − 1 ) .
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