To determine which choice is equivalent to the fraction [tex]\(\frac{1}{\sqrt{x} - \sqrt{x-1}}\)[/tex] when [tex]\(x \geq 1\)[/tex], we can rationalize the denominator. Here are the detailed steps:
1. Given Expression:
[tex]\[
\frac{1}{\sqrt{x} - \sqrt{x-1}}
\][/tex]
2. Multiply by the Conjugate:
We multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{x} + \sqrt{x-1}\)[/tex], to rationalize it.
[tex]\[
\frac{1}{\sqrt{x} - \sqrt{x-1}} \cdot \frac{\sqrt{x} + \sqrt{x-1}}{\sqrt{x} + \sqrt{x-1}}
\][/tex]
3. Simplify the Denominator:
[tex]\[
(\sqrt{x} - \sqrt{x-1})(\sqrt{x} + \sqrt{x-1}) = (\sqrt{x})^2 - (\sqrt{x-1})^2 = x - (x-1) = x - x + 1 = 1
\][/tex]
4. Simplify the Expression:
Since the denominator becomes 1, the numerator remains:
[tex]\[
\frac{\sqrt{x} + \sqrt{x-1}}{1} = \sqrt{x} + \sqrt{x-1}
\][/tex]
Therefore, the fraction simplifies to [tex]\(\sqrt{x} + \sqrt{x-1}\)[/tex].
The correct equivalent expression is [tex]\(\sqrt{x} + \sqrt{x-1}\)[/tex], which corresponds to option (D):
[tex]\[
\boxed{\sqrt{x}+\sqrt{x-1}}
\][/tex]
