For the problem at hand, we need to find the location of point [tex]\( R \)[/tex] on the number line. Point [tex]\( R \)[/tex] partitions the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a [tex]\( 3:2 \)[/tex] ratio.
The section formula for a point dividing a line segment internally in a given ratio is given by:
[tex]\[ R = \frac{m x_2 + n x_1}{m + n} \][/tex]
Given:
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex]
- [tex]\( x_1 = -2 \)[/tex] (coordinate of point [tex]\( Q \)[/tex])
- [tex]\( x_2 = 6 \)[/tex] (coordinate of point [tex]\( S \)[/tex])
Let's substitute these values into the section formula:
[tex]\[ R = \frac{3 \cdot 6 + 2 \cdot (-2)}{3 + 2} \][/tex]
Now, simplify this step-by-step:
1. Compute the products in the numerator:
[tex]\[ 3 \cdot 6 = 18 \][/tex]
[tex]\[ 2 \cdot (-2) = -4 \][/tex]
2. Substitute these results back into the formula:
[tex]\[ R = \frac{18 + (-4)}{5} \][/tex]
3. Simplify the expression inside the numerator:
[tex]\[ 18 + (-4) = 14 \][/tex]
4. Divide by the denominator:
[tex]\[ R = \frac{14}{5} \][/tex]
Thus, the location of point [tex]\( R \)[/tex] on the number line is:
[tex]\[ \frac{14}{5} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\frac{14}{5}} \][/tex]
