Subtract the following. [tex]$\frac{3 x+1}{x}-\frac{3 x}{x+2}$[/tex]
Answer (2)
Find a common denominator: x ( x + 2 ) .
Rewrite each fraction with the common denominator.
Combine the fractions and expand the numerator.
Simplify the numerator to get the final expression: x ( x + 2 ) 7 x + 2 .
Explanation
Problem Analysis We are asked to subtract two rational expressions: x 3 x + 1 − x + 2 3 x . To do this, we need to find a common denominator and combine the fractions.
Finding Common Denominator The common denominator for the two fractions is x ( x + 2 ) . We rewrite each fraction with this common denominator: x 3 x + 1 − x + 2 3 x = x ( x + 2 ) ( 3 x + 1 ) ( x + 2 ) − x ( x + 2 ) 3 x ( x )
Combining Fractions Now we combine the fractions: x ( x + 2 ) ( 3 x + 1 ) ( x + 2 ) − 3 x 2
Expanding Numerator Next, we expand the numerator: x ( x + 2 ) 3 x 2 + 6 x + x + 2 − 3 x 2
Simplifying Numerator Finally, we simplify the numerator: x ( x + 2 ) 7 x + 2
Final Answer The simplified expression is x ( x + 2 ) 7 x + 2 .
Examples
Rational expressions are useful in many areas of science and engineering. For example, in electrical engineering, the impedance of a circuit can be represented as a rational expression. Simplifying such expressions can help engineers analyze and design circuits more efficiently. Also, in physics, rational expressions can appear when dealing with lenses and optics, where simplifying these expressions can help determine the focal length and image distance.
To subtract the rational expressions x 3 x + 1 and x + 2 3 x , we first find a common denominator, rewrite the fractions, and then combine them. The final simplified result is x ( x + 2 ) 7 x + 2 .
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