The problem requires finding the length of $GH$ on a line segment. Here’s the breakdown of the information given:

• $FG = 21$
• $GH = 2x + 15$
• $FH = x + 32$

According to the segment addition postulate, the sum of the segments $FG$ and $GH$ equals the entire segment $FH$: $FG + GH = FH$ Substituting the given values: $21 + (2x + 15) = x + 32$

Now, let's solve for $x$:

1. Combine like terms: $21 + 2x + 15 = x + 32$ $2x + 36 = x + 32$

2. Subtract $x$ from both sides: $x + 36 = 32$

3. Subtract 36 from both sides: $x = -4$

Now that we have $x = -4$, let's find $GH$: $GH = 2x + 15 = 2(-4) + 15 = -8 + 15 = 7$

So, the length of $GH$ is $7$.

Would you like a detailed explanation or have any other questions? Here are some related questions you might find helpful:

1. How do you solve equations involving line segments?
2. What is the segment addition postulate?
3. How can negative values affect segment lengths in geometry problems?
4. What are the properties of line segments in coordinate geometry?
5. How to verify the solution of a geometry problem?

Tip: Always verify the calculated segment lengths by substituting back into the original equation to ensure consistency.