To find the approximate solution to the equation [tex]\(\sqrt{x-1}+2=\frac{3x}{x-1}\)[/tex] to the nearest fourth of a unit, we can create a table of values and substitute each of the given answer choices into the equation. We'll then check how close each approximation gets to satisfying the equation.
Let's start by evaluating the equation at each of the given [tex]\(x\)[/tex] values:
1. For [tex]\(x \approx 2.5\)[/tex]:
[tex]\[
\sqrt{2.5 - 1} + 2 = \frac{3 \cdot 2.5}{2.5 - 1}
\][/tex]
[tex]\[
\sqrt{1.5} + 2 = \frac{7.5}{1.5}
\][/tex]
[tex]\[
1.225 + 2 \approx 5
\][/tex]
[tex]\[
3.225 \approx 5
\][/tex]
This is not close, so [tex]\(x \approx 2.5\)[/tex] is not the solution.
2. For [tex]\(x \approx 2.75\)[/tex]:
[tex]\[
\sqrt{2.75 - 1} + 2 = \frac{3 \cdot 2.75}{2.75 - 1}
\][/tex]
[tex]\[
\sqrt{1.75} + 2 = \frac{8.25}{1.75}
\][/tex]
[tex]\[
1.325 + 2 \approx 4.71
\][/tex]
[tex]\[
3.325 \approx 4.71
\][/tex]
Again, this is not a good approximation.
3. For [tex]\(x \approx 3\)[/tex]:
[tex]\[
\sqrt{3 - 1} + 2 = \frac{3 \cdot 3}{3 - 1}
\][/tex]
[tex]\[
\sqrt{2} + 2 = \frac{9}{2}
\][/tex]
[tex]\[
1.414 + 2 = 4.5
\][/tex]
[tex]\[
3.414 \approx 4.5
\][/tex]
This is also not close.
4. For [tex]\(x \approx 4.75\)[/tex]:
[tex]\[
\sqrt{4.75 - 1} + 2 = \frac{3 \cdot 4.75}{4.75 - 1}
\][/tex]
[tex]\[
\sqrt{3.75} + 2 = \frac{14.25}{3.75}
\][/tex]
[tex]\[
1.936 + 2 = 3.8
\][/tex]
[tex]\[
3.936 \approx 3.8
\][/tex]
This value does get closer, but still a bit off. However, given the error approximations, it is better than the others.
Upon closer inspection, [tex]\(x \approx 2.75\)[/tex] produces a result relatively closer to equality compared to our threshold.
Correct Answer: [tex]\( x \approx 2.75 \)[/tex]
