To find the location of point [tex]\(R\)[/tex] on the number line, we need to correctly apply the section formula given the ratio and endpoints. Here's the step-by-step solution:
1. Identify the given points and the ratio:
- The endpoint [tex]\(Q\)[/tex] is at [tex]\(-2\)[/tex],
- The endpoint [tex]\(S\)[/tex] is at [tex]\(6\)[/tex],
- The ratio in which [tex]\(R\)[/tex] partitions the segment from [tex]\(Q\)[/tex] to [tex]\(S\)[/tex] is [tex]\(3:2\)[/tex].
2. Let [tex]\(m = 3\)[/tex] and [tex]\(n = 2\)[/tex] represent the parts of the ratio. Let [tex]\(x_1 = -2\)[/tex] (coordinate of [tex]\(Q\)[/tex]) and [tex]\(x_2 = 6\)[/tex] (coordinate of [tex]\(S\)[/tex]).
3. Use the section formula:
[tex]\[
R = \frac{m x_2 + n x_1}{m + n}
\][/tex]
4. Substitute the given values into the formula:
[tex]\[
R = \frac{3(6) + 2(-2)}{3 + 2}
\][/tex]
5. Perform the multiplications inside the numerator:
[tex]\[
R = \frac{18 + (-4)}{5}
\][/tex]
6. Simplify the numerator:
[tex]\[
R = \frac{14}{5}
\][/tex]
Thus, the location of point [tex]\(R\)[/tex] on the number line is [tex]\(\frac{14}{5}\)[/tex].
Hence, the correct answer is [tex]\(\frac{14}{5}\)[/tex].
