Given a number line with endpoints [tex]\( Q \)[/tex] at [tex]\(-2\)[/tex] and [tex]\( S \)[/tex] at [tex]\( 6 \)[/tex], we are to find the location of point [tex]\( R \)[/tex] that partitions the directed line segment [tex]\( QS \)[/tex] in a [tex]\( 3:2 \)[/tex] ratio.
Let's use the section formula to determine [tex]\( R \)[/tex].
#### Step-by-Step Solution:
1. Assign the given values:
[tex]\[
Q \implies x_1 = -2 \quad \text{and} \quad S \implies x_2 = 6
\][/tex]
The ratio given is [tex]\( 3:2 \)[/tex].
[tex]\[
\Rightarrow m = 3 \quad \text{and} \quad n = 2
\][/tex]
2. The section formula for a point [tex]\( R \)[/tex] that divides the segment [tex]\( QS \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
R = \frac{m x_2 + n x_1}{m+n}
\][/tex]
3. Substituting the values into the section formula:
[tex]\[
R = \frac{3 \cdot 6 + 2 \cdot (-2)}{3 + 2}
\][/tex]
4. Simplify the expression step-by-step:
[tex]\[
R = \frac{18 + (-4)}{5}
\][/tex]
[tex]\[
R = \frac{18 - 4}{5}
\][/tex]
[tex]\[
R = \frac{14}{5}
\][/tex]
Therefore, the location of point [tex]\( R \)[/tex] on the number line is:
[tex]\[
R = \frac{14}{5}
\][/tex]
Among the options given, the correct location of point [tex]\( R \)[/tex] is:
[tex]\[
\boxed{\frac{14}{5}}
\][/tex]
