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Answer :
Final answer:
The simplified expression for 3/x + 2/(x+2) / 3/(x+2) - 2/x is (5x+6)/x - 4.
Explanation:
To simplify the given expression, we first need to find a common denominator for the fractions.
The common denominator for the fractions 3/x, 2/(x+2), 3/(x+2), and 2/x is x(x+2). Then, we rewrite each fraction with the common denominator.
After obtaining a common denominator, we can combine the fractions and simplify. The simplified expression after combining like terms is (5x+6)/x - 4.
To find the common denominator, we multiply each fraction by the missing factor needed to make the denominator x(x+2).
For the first fraction 3/x, we multiply the numerator and denominator by (x+2) to get 3(x+2)/(x(x+2)).
For the second fraction 2/(x+2), the denominator is already (x+2), so we multiply the numerator and denominator by x to get 2x/(x(x+2)).
Similarly, for the third fraction 3/(x+2), we multiply the numerator and denominator by x to get 3x/(x(x+2)). For the fourth fraction 2/x, we multiply the numerator and denominator by (x+2) to get 2(x+2)/(x(x+2)).
After obtaining a common denominator, we combine the fractions by subtracting the numerators and placing the result over the common denominator.
This gives us (3(x+2) + 2x - 3x - 2(x+2)) / (x(x+2)). Simplifying the numerator gives us (5x+6), and the denominator remains x(x+2). Therefore, the simplified expression is (5x+6)/x - 4.
The final answer matches option 3: (5x+6)/x - 4.
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