Answer :
To solve the equation [tex]\( 2 \sqrt{x-1} + 2 = \frac{3x}{x-1} \)[/tex], we will approximate the solution to the nearest fourth of a unit by evaluating the left-hand side and the right-hand side for the given options and seeing which value makes both sides of the equation approximately equal.
Let's consider each of the options:
### Option A: [tex]\( x \approx 2.75 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{2.75 - 1} + 2 = 2 \sqrt{1.75} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{1.75} \approx 1.322 \][/tex]
[tex]\[ 2 \cdot 1.322 + 2 \approx 2.644 + 2 = 4.644 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 2.75}{2.75 - 1} = \frac{8.25}{1.75} \][/tex]
Approximating the division:
[tex]\[ \frac{8.25}{1.75} \approx 4.714 \][/tex]
3. Comparing both sides:
[tex]\[ 4.644 \approx 4.714 \ (close) \][/tex]
### Option B: [tex]\( x \approx 2.5 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{2.5 - 1} + 2 = 2 \sqrt{1.5} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{1.5} \approx 1.225 \][/tex]
[tex]\[ 2 \cdot 1.225 + 2 \approx 2.450 + 2 = 4.450 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 2.5}{2.5 - 1} = \frac{7.5}{1.5} \][/tex]
Approximating the division:
[tex]\[ \frac{7.5}{1.5} = 5 \][/tex]
3. Comparing both sides:
[tex]\[ 4.450 \not\approx 5 \ (not close) \][/tex]
### Option C: [tex]\( x \approx 4.75 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{4.75 - 1} + 2 = 2 \sqrt{3.75} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{3.75} \approx 1.936 \][/tex]
[tex]\[ 2 \cdot 1.936 + 2 \approx 3.872 + 2 = 5.872 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 4.75}{4.75 - 1} = \frac{14.25}{3.75} \][/tex]
Approximating the division:
[tex]\[ \frac{14.25}{3.75} \approx 3.8 \][/tex]
3. Comparing both sides:
[tex]\[ 5.872 \not\approx 3.8 \ (not close) \][/tex]
### Option D: [tex]\( x \approx 3 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{3 - 1} + 2 = 2 \sqrt{2} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{2} \approx 1.414 \][/tex]
[tex]\[ 2 \cdot 1.414 + 2 \approx 2.828 + 2 = 4.828 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 3}{3 - 1} = \frac{9}{2} \][/tex]
Approximating the division:
[tex]\[ \frac{9}{2} = 4.5 \][/tex]
3. Comparing both sides:
[tex]\[ 4.828 \approx 4.5 \ (close but slightly off) \][/tex]
### Conclusion
After evaluating all the options, the values at [tex]\( x \approx 2.75 \)[/tex] yield results that are closest on both sides of the equation. Therefore, the correct approximation to the nearest fourth of a unit is:
[tex]\[ \boxed{x \approx 2.75} \][/tex]
Let's consider each of the options:
### Option A: [tex]\( x \approx 2.75 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{2.75 - 1} + 2 = 2 \sqrt{1.75} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{1.75} \approx 1.322 \][/tex]
[tex]\[ 2 \cdot 1.322 + 2 \approx 2.644 + 2 = 4.644 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 2.75}{2.75 - 1} = \frac{8.25}{1.75} \][/tex]
Approximating the division:
[tex]\[ \frac{8.25}{1.75} \approx 4.714 \][/tex]
3. Comparing both sides:
[tex]\[ 4.644 \approx 4.714 \ (close) \][/tex]
### Option B: [tex]\( x \approx 2.5 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{2.5 - 1} + 2 = 2 \sqrt{1.5} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{1.5} \approx 1.225 \][/tex]
[tex]\[ 2 \cdot 1.225 + 2 \approx 2.450 + 2 = 4.450 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 2.5}{2.5 - 1} = \frac{7.5}{1.5} \][/tex]
Approximating the division:
[tex]\[ \frac{7.5}{1.5} = 5 \][/tex]
3. Comparing both sides:
[tex]\[ 4.450 \not\approx 5 \ (not close) \][/tex]
### Option C: [tex]\( x \approx 4.75 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{4.75 - 1} + 2 = 2 \sqrt{3.75} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{3.75} \approx 1.936 \][/tex]
[tex]\[ 2 \cdot 1.936 + 2 \approx 3.872 + 2 = 5.872 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 4.75}{4.75 - 1} = \frac{14.25}{3.75} \][/tex]
Approximating the division:
[tex]\[ \frac{14.25}{3.75} \approx 3.8 \][/tex]
3. Comparing both sides:
[tex]\[ 5.872 \not\approx 3.8 \ (not close) \][/tex]
### Option D: [tex]\( x \approx 3 \)[/tex]
1. Calculate the left side:
[tex]\[ 2 \sqrt{3 - 1} + 2 = 2 \sqrt{2} + 2 \][/tex]
Approximating the square root:
[tex]\[ \sqrt{2} \approx 1.414 \][/tex]
[tex]\[ 2 \cdot 1.414 + 2 \approx 2.828 + 2 = 4.828 \][/tex]
2. Calculate the right side:
[tex]\[ \frac{3 \cdot 3}{3 - 1} = \frac{9}{2} \][/tex]
Approximating the division:
[tex]\[ \frac{9}{2} = 4.5 \][/tex]
3. Comparing both sides:
[tex]\[ 4.828 \approx 4.5 \ (close but slightly off) \][/tex]
### Conclusion
After evaluating all the options, the values at [tex]\( x \approx 2.75 \)[/tex] yield results that are closest on both sides of the equation. Therefore, the correct approximation to the nearest fourth of a unit is:
[tex]\[ \boxed{x \approx 2.75} \][/tex]
