To solve the equation [tex]\(2 \sqrt{x-1}+2=\frac{3 x}{x-1}\)[/tex] using a table of values and approximate the solution to the nearest fourth of a unit, we follow the steps below:
1. Understand the equation:
The equation we need to solve is:
[tex]\[
2 \sqrt{x-1} + 2 = \frac{3x}{x-1}
\][/tex]
2. Create a table of values:
We'll evaluate both sides of the equation for various values of [tex]\(x\)[/tex] near the given options and see which value makes both sides approximately equal.
3. Evaluate both sides of the equation for the given options:
Let's start by evaluating both sides for each of the options:
- For [tex]\(x = 2.5\)[/tex]:
[tex]\[
\text{Left side: } 2 \sqrt{2.5 - 1} + 2 = 2 \sqrt{1.5} + 2 \approx 2 \times 1.225 + 2 \approx 2.45 + 2 = 4.45
\][/tex]
[tex]\[
\text{Right side: } \frac{3 \times 2.5}{2.5 - 1} = \frac{7.5}{1.5} = 5
\][/tex]
- For [tex]\(x = 4.75\)[/tex]:
[tex]\[
\text{Left side: } 2 \sqrt{4.75 - 1} + 2 = 2 \sqrt{3.75} + 2 \approx 2 \times 1.936 + 2 \approx 3.872 + 2 = 5.872
\][/tex]
[tex]\[
\text{Right side: } \frac{3 \times 4.75}{4.75 - 1} = \frac{14.25}{3.75} \approx 3.8
\][/tex]
- For [tex]\(x = 3\)[/tex]:
[tex]\[
\text{Left side: } 2 \sqrt{3 - 1} + 2 = 2 \sqrt{2} + 2 \approx 2 \times 1.414 + 2 \approx 2.828 + 2 = 4.828
\][/tex]
[tex]\[
\text{Right side: } \frac{3 \times 3}{3 - 1} = \frac{9}{2} = 4.5
\][/tex]
- For [tex]\(x = 2.75\)[/tex]:
[tex]\[
\text{Left side: } 2 \sqrt{2.75 - 1} + 2 = 2 \sqrt{1.75} + 2 \approx 2 \times 1.323 + 2 \approx 2.646 + 2 = 4.646
\][/tex]
[tex]\[
\text{Right side: } \frac{3 \times 2.75}{2.75 - 1} = \frac{8.25}{1.75} \approx 4.714
\][/tex]
4. Compare the values:
- For [tex]\(x = 2.5\)[/tex], the values do not match closely: [tex]\(4.45 \neq 5\)[/tex].
- For [tex]\(x = 4.75\)[/tex], the values do not match closely: [tex]\(5.872 \neq 3.8\)[/tex].
- For [tex]\(x = 3\)[/tex], the values are quite close: [tex]\(4.828 \approx 4.5\)[/tex].
- For [tex]\(x = 2.75\)[/tex], the values are close: [tex]\(4.646 \approx 4.714\)[/tex].
5. Select the closest match:
Between [tex]\(x = 3\)[/tex] and [tex]\(x = 2.75\)[/tex], the closest match is [tex]\(x = 3\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{x \approx 3} \][/tex]
