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Simplifying the Expression (x+1/2)/(2+1/x)
This article will guide you through simplifying the expression (x+1/2)/(2+1/x). We'll break down the steps and explain the rationale behind each one.
1. Combining Terms in the Numerator and Denominator
The first step is to combine the terms in both the numerator and denominator. We can do this by finding a common denominator for each fraction:
- Numerator: (x + 1/2) = (2x/2 + 1/2) = (2x+1)/2
- Denominator: (2 + 1/x) = (2x/x + 1/x) = (2x+1)/x
Now our expression looks like this: ((2x+1)/2) / ((2x+1)/x)
2. Dividing Fractions
Dividing by a fraction is the same as multiplying by its reciprocal. This means we can flip the denominator and multiply:
((2x+1)/2) * (x/(2x+1))
3. Cancellation and Simplification
Notice that (2x+1) appears in both the numerator and denominator. We can cancel these terms out:
(1/2) * (x/1)
Finally, we can simplify to get our final answer:
x/2
Conclusion
Therefore, the simplified expression of (x+1/2)/(2+1/x) is x/2. Remember, this simplification is only valid for values of x that don't make the original expression undefined (i.e., x cannot be 0 or -1/2).