Point Q is on line segment start overline, P, R, end overline
PR
. Given Q, R, equals, 2, x, plus, 4, commaQR=2x+4, P, Q, equals, x, commaPQ=x, and P, R, equals, 4, x, minus, 10, commaPR=4x−10, determine the numerical length of start overline, P, R, end overline, .
PR

Given that point \( Q \) is on line segment \( \overline{PR} \), we have the following information:

- \( QR = 2x + 4 \)
- \( PQ = x \)
- \( PR = 4x - 10 \)

According to the segment addition postulate, the sum of the lengths of segments \( PQ \) and \( QR \) should equal the length of the entire segment \( PR \). Therefore, we can write:

\[
PQ + QR = PR
\]

Substituting the given expressions:

\[
x + (2x + 4) = 4x - 10
\]

Simplifying the left-hand side:

\[
3x + 4 = 4x - 10
\]

Next, we'll solve for \( x \). Subtract \( 3x \) from both sides:

\[
4 = x - 10
\]

Add 10 to both sides:

\[
x = 14
\]

Now that we have \( x = 14 \), we can find the length of \( PR \) by substituting \( x \) back into the expression for \( PR \):

\[
PR = 4x - 10 = 4(14) - 10 = 56 - 10 = 46
\]

Thus, the numerical length of \( \overline{PR} \) is \( \boxed{46} \).

Would you like more details on any step? Here are some related questions:

1. How do you apply the segment addition postulate in similar problems?
2. What if point Q was not between P and R? How would the approach change?
3. How do you check for consistency in geometric problems involving algebra?
4. How can you verify that your solution is correct for line segment problems?
5. What are common pitfalls when solving for variables in geometric contexts?

**Tip:** Always substitute back the value of the variable into the original equation to verify your solution.

Learn how to determine the length of line segment PR when point Q lies on the segment. This problem involves applying the segment addition postulate and solving algebraic equations. Suitable for high school level students.

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