To solve the equation [tex]\( 2 \sqrt{x - 1} + 2 = \frac{3x}{x - 1} \)[/tex] and approximate the solution to the nearest fourth of a unit, we can evaluate the left side and the right side of the equation at specific [tex]\( x \)[/tex] values and find the one where the two sides are closest to being equal. The absolute difference between both sides should be minimized.
We evaluate the equation at the following values: [tex]\( x = 4.75 \)[/tex], [tex]\( x = 2.5 \)[/tex], [tex]\( x = 2.75 \)[/tex], and [tex]\( x = 3 \)[/tex]. We'll compare [tex]\( 2 \sqrt{x - 1} + 2 \)[/tex] with [tex]\( \frac{3x}{x - 1} \)[/tex].
Let's start with these values:
1. For [tex]\( x = 4.75 \)[/tex]:
- Left side: [tex]\( 2 \sqrt{4.75 - 1} + 2 \)[/tex]
- Right side: [tex]\( \frac{3 \times 4.75}{4.75 - 1} \)[/tex]
- Difference: [tex]\( \left| 2 \sqrt{4.75 - 1} + 2 - \frac{3 \times 4.75}{4.75 - 1} \right| \approx 2.072983346207417 \)[/tex]
2. For [tex]\( x = 2.5 \)[/tex]:
- Left side: [tex]\( 2 \sqrt{2.5 - 1} + 2 \)[/tex]
- Right side: [tex]\( \frac{3 \times 2.5}{2.5 - 1} \)[/tex]
- Difference: [tex]\( \left| 2 \sqrt{2.5 - 1} + 2 - \frac{3 \times 2.5}{2.5 - 1} \right| \approx -0.5505102572168221 \)[/tex]
3. For [tex]\( x = 2.75 \)[/tex]:
- Left side: [tex]\( 2 \sqrt{2.75 - 1} + 2 \)[/tex]
- Right side: [tex]\( \frac{3 \times 2.75}{2.75 - 1} \)[/tex]
- Difference: [tex]\( \left| 2 \sqrt{2.75 - 1} + 2 - \frac{3 \times 2.75}{2.75 - 1} \right| \approx -0.0685344032211237 \)[/tex]
4. For [tex]\( x = 3 \)[/tex]:
- Left side: [tex]\( 2 \sqrt{3 - 1} + 2 \)[/tex]
- Right side: [tex]\( \frac{3 \times 3}{3 - 1} \)[/tex]
- Difference: [tex]\( \left| 2 \sqrt{3 - 1} + 2 - \frac{3 \times 3}{3 - 1} \right| \approx 0.32842712474618985 \)[/tex]
The closest approximation is given by the smallest absolute difference. Here, the smallest difference is approximately [tex]\( -0.0685344032211237 \)[/tex] when [tex]\( x = 2.75 \)[/tex].
Thus, the correct answer is:
C. [tex]\( x \approx 2.75 \)[/tex]
