Explore a world of knowledge and get your questions answered on IDNLearn.com. Discover prompt and accurate answers from our community of experienced professionals.

Select the correct answer.

Using a table of values, approximate the solution to the equation below to the nearest fourth of a unit.

[tex]\[
2 \sqrt{x-1}+2=\frac{3 x}{x-1}
\][/tex]

A. [tex]\( x \approx 4.75 \)[/tex]
B. [tex]\( x \approx 2.75 \)[/tex]
C. [tex]\( x \approx 3 \)[/tex]
D. [tex]\( x \approx 2.5 \)[/tex]


Sagot :

To solve the equation [tex]\(2 \sqrt{x-1} + 2 = \frac{3x}{x-1}\)[/tex], we can approximate the solution by evaluating the expression on both sides of the equation for various values of [tex]\(x\)[/tex]. Let's construct a table of values for the chosen [tex]\(x\)[/tex] values: 2.5, 2.75, 3, and 4.75, and find which one is closest to satisfying the equation.

First, let's plug in each value into the left-hand side [tex]\(2 \sqrt{x-1} + 2\)[/tex] and the right-hand side [tex]\(\frac{3x}{x-1}\)[/tex]:

1. For [tex]\(x = 2.5\)[/tex]:
[tex]\[ \text{LHS: } 2 \sqrt{2.5-1} + 2 = 2 \sqrt{1.5} + 2 \approx 2 \times 1.2247 + 2 \approx 2.4494 + 2 = 4.4494 \][/tex]
[tex]\[ \text{RHS: } \frac{3 \cdot 2.5}{2.5 - 1} = \frac{7.5}{1.5} = 5 \][/tex]
[tex]\[ \text{Difference: } 4.4494 - 5 \approx -0.5505 \][/tex]

2. For [tex]\(x = 2.75\)[/tex]:
[tex]\[ \text{LHS: } 2 \sqrt{2.75-1} + 2 = 2 \sqrt{1.75} + 2 \approx 2 \times 1.3229 + 2 \approx 2.6458 + 2 = 4.6458 \][/tex]
[tex]\[ \text{RHS: } \frac{3 \cdot 2.75}{2.75 - 1} = \frac{8.25}{1.75} \approx 4.7143 \][/tex]
[tex]\[ \text{Difference: } 4.6458 - 4.7143 \approx -0.0685 \][/tex]

3. For [tex]\(x = 3\)[/tex]:
[tex]\[ \text{LHS: } 2 \sqrt{3-1} + 2 = 2 \sqrt{2} + 2 \approx 2 \times 1.4142 + 2 = 2.8284 + 2 = 4.8284 \][/tex]
[tex]\[ \text{RHS: } \frac{3 \cdot 3}{3 - 1} = \frac{9}{2} = 4.5 \][/tex]
[tex]\[ \text{Difference: } 4.8284 - 4.5 \approx 0.3284 \][/tex]

4. For [tex]\(x = 4.75\)[/tex]:
[tex]\[ \text{LHS: } 2 \sqrt{4.75-1} + 2 = 2 \sqrt{3.75} + 2 \approx 2 \times 1.9365 + 2 = 3.8730 + 2 = 5.8730 \][/tex]
[tex]\[ \text{RHS: } \frac{3 \cdot 4.75}{4.75 - 1} = \frac{14.25}{3.75} \approx 3.80 \][/tex]
[tex]\[ \text{Difference: } 5.8730 - 3.80 = 2.073 \][/tex]

From these calculations, we observe that the smallest difference in magnitude is for [tex]\(x = 2.75\)[/tex], with a value of approximately [tex]\(-0.0685\)[/tex].

Therefore, the closest approximation to the solution of the equation [tex]\(2 \sqrt{x-1} + 2 = \frac{3x}{x-1}\)[/tex] to the nearest fourth of a unit is:

B. [tex]\(x \approx 2.75\)[/tex]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.