To determine the location of point [tex]\( R \)[/tex] on the number line, we can use the section formula for a line segment divided in a given ratio. In this problem, point [tex]\( R \)[/tex] divides the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a ratio of 3:2. Let's denote the given values:
- The coordinates of point [tex]\( Q \)[/tex] are [tex]\( x_1 = -2 \)[/tex].
- The coordinates of point [tex]\( S \)[/tex] are [tex]\( x_2 = 6 \)[/tex].
- The ratio [tex]\( m:n = 3:2 \)[/tex].
The section formula for a point dividing a line segment in a given ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
R = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
Substituting the given values into the formula:
[tex]\[
R = \frac{3 \cdot 6 + 2 \cdot (-2)}{3 + 2}
\][/tex]
Next, we calculate the numerator and the denominator:
[tex]\[
R = \frac{18 + (-4)}{5}
\][/tex]
Simplify the expression inside the fraction:
[tex]\[
R = \frac{18 - 4}{5}
\][/tex]
[tex]\[
R = \frac{14}{5}
\][/tex]
Thus, the location of point [tex]\( R \)[/tex] on the number line is:
[tex]\[
\boxed{\frac{14}{5}}
\][/tex]
This confirms that point [tex]\( R \)[/tex] is located at [tex]\( \frac{14}{5} \)[/tex] on the number line.
