Question: If point $h$ is between point $g$ and point $I$ and if $g h$ equals $2x - 9$, $h I$ equals $14$ and $g I$ equals $3x - 8$, then find $g I$
1. **State the problem:** We are given three points $g$, $h$, and $I$ on a line with $h$ between $g$ and $I$. The distances are $g h = 2x - 9$, $h I = 14$, and $g I = 3x - 8$. We need to find the length $g I$.
2. **Use the segment addition postulate:** Since $h$ is between $g$ and $I$, the total length $g I$ is the sum of $g h$ and $h I$:
$$g I = g h + h I$$
3. **Substitute the given expressions:**
$$3x - 8 = (2x - 9) + 14$$
4. **Simplify the right side:**
$$3x - 8 = 2x - 9 + 14$$
$$3x - 8 = 2x + 5$$
5. **Isolate $x$ by subtracting $2x$ from both sides:**
$$3x - 2x - 8 = 2x - 2x + 5$$
$$x - 8 = 5$$
6. **Add $8$ to both sides:**
$$x - 8 + 8 = 5 + 8$$
$$x = 13$$
7. **Find $g I$ by substituting $x = 13$ into $3x - 8$:**
$$g I = 3(13) - 8 = 39 - 8 = 31$$
**Final answer:**
$$\boxed{31}$$