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Let's solve the problems step-by-step.

### Problem 6: Find PR

The points \(P\), \(Q\), and \(R\) lie on a line segment, where:

- \(PQ = 8\)
- \(QR = 2x - 15\)
- \(PR = PQ + QR = 5 + x\)

We need to set up an equation where the sum of \(PQ\) and \(QR\) equals \(PR\):

\[
PQ + QR = PR
\]

Substituting the given values:

\[
8 + (2x - 15) = 5 + x
\]

Simplify and solve for \(x\):

\[
8 + 2x - 15 = 5 + x
\]
\[
2x - 7 = 5 + x
\]
\[
2x - x = 5 + 7
\]
\[
x = 12
\]

Now, substitute \(x = 12\) back into the expressions for \(QR\) and \(PR\):

\[
QR = 2x - 15 = 2(12) - 15 = 24 - 15 = 9
\]
\[
PR = 5 + x = 5 + 12 = 17
\]

### Answer for Problem 6:
\[
PR = 17
\]

### Problem 8: Find RT

The points \(R\), \(S\), and \(T\) lie on another line segment, where:

- \(RS = x + 2\)
- \(ST = 4\)
- \(RT = RS + ST = 2x - 4\)

Set up an equation:

\[
RS + ST = RT
\]

Substitute the given values:

\[
(x + 2) + 4 = 2x - 4
\]

Simplify and solve for \(x\):

\[
x + 2 + 4 = 2x - 4
\]
\[
x + 6 = 2x - 4
\]
\[
6 + 4 = 2x - x
\]
\[
10 = x
\]

Substitute \(x = 10\) back into the expressions for \(RS\) and \(RT\):

\[
RS = x + 2 = 10 + 2 = 12
\]
\[
RT = 2x - 4 = 2(10) - 4 = 20 - 4 = 16
\]

### Answer for Problem 8:
\[
RT = 16
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How would the results change if the values of \(PQ\), \(QR\), or \(ST\) were different?
2. What are the possible values of \(x\) for which these equations have valid geometric interpretations?
3. How can you verify the calculated lengths using a geometric approach?
4. How do the equations change if the segments are not collinear?
5. How would you approach solving for multiple variables if more segments were involved?

### Tip:
When solving problems with line segments, always ensure that all segments align properly and that their sums or differences make sense geometrically!

Find the lengths of PR and RT from the provided diagram where multiple points and distances are given with variables.

Learn how to find the lengths of segments PR and RT using algebraic expressions and the segment addition postulate. This guide provides step-by-step solutions to solve for variable lengths in a geometric context, ideal for high school students studying algebra.

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