Alright, let's solve this step by step.
1. Identify the coordinates of points F and G:
- Point F is at 4.
- Point G is at -2.
2. Determine the given ratio FH:HG:
- The ratio of FH to HG is given as 3:9.
3. Simplify the ratio:
- The ratio 3:9 can be simplified to 1:3.
4. Set up the distances based on the simplified ratio:
- Let the distance FH be [tex]\( x \)[/tex].
- Then the distance HG is [tex]\( 3x \)[/tex].
5. Express the total distance from F to G:
- The total distance from F to G is the sum of FH and HG, which is [tex]\( FH + HG = x + 3x = 4x \)[/tex].
6. Calculate the absolute distance between F and G:
- The distance between F and G is the absolute difference of their coordinates:
[tex]\[ \text{distance} = |4 - (-2)| = 6. \][/tex]
7. Solve for [tex]\( x \)[/tex]:
- Since [tex]\( 4x = 6 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{6}{4} = 1.5. \][/tex]
8. Find the position of H:
- The position of H can be found by subtracting the distance FH from F:
[tex]\[ H = F - FH = 4 - 1.5 = 2.5. \][/tex]
Hence, point H lies at [tex]\( 2.5 \)[/tex] on the number line.
