Perform the indicated operation. Simplify if possible. [tex]\frac{9 x}{x+9}-\frac{4 x-45}{x+9}[/tex]
Answer (2)
Subtract the numerators: x + 9 9 x − ( 4 x − 45 ) .
Distribute the negative sign: x + 9 9 x − 4 x + 45 .
Combine like terms: x + 9 5 x + 45 .
Factor and cancel: x + 9 5 ( x + 9 ) = 5 .
The simplified expression is 5 .
Explanation
Understanding the Problem We are asked to subtract two rational expressions with the same denominator. The expressions are x + 9 9 x and x + 9 4 x − 45 .
Subtracting Numerators Since the denominators are the same, we can subtract the numerators: x + 9 9 x − ( 4 x − 45 )
Distributing the Negative Sign Distribute the negative sign in the numerator: x + 9 9 x − 4 x + 45
Combining Like Terms Combine like terms in the numerator: x + 9 5 x + 45
Factoring the Numerator Factor the numerator: x + 9 5 ( x + 9 )
Canceling Common Factors Cancel the common factor of ( x + 9 ) from the numerator and denominator: x + 9 5 ( x + 9 ) = 5
Final Answer The simplified expression is 5 .
Examples
Rational expressions are useful in real life for calculating rates, proportions, and changes in quantities. For example, if you are tracking the speed of a car as it accelerates, you might use a rational expression to represent the relationship between distance and time. Simplifying these expressions helps in making quick and accurate calculations, which is essential in fields like engineering, physics, and economics.
The expression x + 9 9 x − x + 9 4 x − 45 simplifies to 5 by subtracting the numerators and cancelling common factors in the denominator. This involves distributing the negative sign and combining like terms. The final answer is 5 .
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