Alright, let's solve this step by step:
1. Identify the coordinates of points [tex]\( D \)[/tex] and [tex]\( E \)[/tex]. We know that [tex]\( D \)[/tex] is at [tex]\(-6\)[/tex] and [tex]\( E \)[/tex] is at [tex]\(8\)[/tex].
2. Calculate the total distance between [tex]\( D \)[/tex] and [tex]\( E \)[/tex]. This distance is:
[tex]\[
E - D = 8 - (-6) = 8 + 6 = 14
\][/tex]
So, the total distance between [tex]\( D \)[/tex] and [tex]\( E \)[/tex] is [tex]\(14\)[/tex].
3. Next, we are given the ratio [tex]\( DF : FE = 3 : 1 \)[/tex]. This means that for every 4 parts of total distance, 3 parts belong to [tex]\( DF \)[/tex] and 1 part belongs to [tex]\( FE \)[/tex].
The total ratio is [tex]\( 3 + 1 = 4 \)[/tex].
4. Calculate the distance [tex]\( DF \)[/tex], which is three parts out of the total 4 parts of distance:
[tex]\[
DF = \left(\frac{3}{4}\right) \times 14 = 10.5
\][/tex]
5. To find the position of point [tex]\( F \)[/tex], add the distance [tex]\( DF \)[/tex] to the starting point [tex]\( D \)[/tex]:
[tex]\[
F = D + DF = -6 + 10.5 = 4.5
\][/tex]
So, point [tex]\( F \)[/tex] is at [tex]\( 4.5 \)[/tex] on the number line.
Therefore, the correct answer to fill in the blank is:
[tex]\[ \boxed{4.5} \][/tex]
Point [tex]\( F \)[/tex] is at [tex]\( 4.5 \)[/tex] on the number line.
