To determine the length of [tex]\( \overline{AB} \)[/tex], we start by analyzing the given points and their relationship on the number line.
1. Determine the length of [tex]\( \overline{AC} \)[/tex]:
- Point [tex]\( A \)[/tex] is at -3.4.
- Point [tex]\( C \)[/tex] is at 0.6.
- The distance between [tex]\( A \)[/tex] and [tex]\( C \)[/tex] can be found by calculating the absolute difference between these two points.
- Therefore, [tex]\( AC = |C - A| = |0.6 - (-3.4)| = |0.6 + 3.4| = 4.0 \)[/tex] units.
2. Understand the ratio [tex]\( \overline{AC} : \overline{CB} = 4 : 7 \)[/tex]:
- Let the length of [tex]\( \overline{AC} \)[/tex] be [tex]\( 4x \)[/tex] and the length of [tex]\( \overline{CB} \)[/tex] be [tex]\( 7x \)[/tex].
- Since we already found [tex]\( \overline{AC} \)[/tex] to be 4.0 units, we can deduce that [tex]\( 4x = 4.0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( x = 1 \)[/tex].
3. Calculate the total length [tex]\( \overline{AB} \)[/tex]:
- Using [tex]\( x = 1 \)[/tex], the length of [tex]\( \overline{CB} \)[/tex] is [tex]\( 7x = 7 \times 1 = 7.0 \)[/tex] units.
- Therefore, the total length [tex]\( \overline{AB} \)[/tex] is [tex]\( \overline{AC} + \overline{CB} = 4.0 + 7.0 = 11.0 \)[/tex] units.
Thus, the correct answer is:
[tex]\[ AB = 11 \text{ units} \][/tex]
So, [tex]\( \boxed{11} \)[/tex] units is the length of [tex]\( \overline{AB} \)[/tex].
