To solve for \( PR \), we can start by using the fact that \( PQ + QR = PR \).
We are given the lengths:
- \( PQ = 20 \)
- \( QR = x + 15 \)
- \( PR = 4x - 1 \)
According to the segment addition postulate, we can write:
\[ PQ + QR = PR \]
Substituting in the lengths we have:
\[ 20 + (x + 15) = 4x - 1 \]
Now, simplify the left side:
\[ 20 + x + 15 = 4x - 1 \]
This becomes:
\[ 35 + x = 4x - 1 \]
Next, let's isolate \( x \) by moving \( x \) to the right side and \( -1 \) to the left side:
\[ 35 + 1 = 4x - x \]
This simplifies to:
\[ 36 = 3x \]
Now, divide both sides by 3:
\[ x = 12 \]
Now that we have \( x \), we can find \( PR \) using the expression for \( PR \):
\[ PR = 4x - 1 \]
Substituting \( x = 12 \):
\[ PR = 4(12) - 1 \]
Calculating:
\[ PR = 48 - 1 = 47 \]
So, \( PR = 47 \).
Thus, the value of \( PR \) is:
\[ \boxed{47} \]