https://file.aiquickdraw.com/mathbot/file/1726492018440p9ox40yn.jpg

To solve the problem:

We are given:
- \( ED = 6x - 4 \)
- \( DF = 3x + 5 \)
- \( EF = 46 \)

Since \( D \) is between \( E \) and \( F \), we know that:
\[
ED + DF = EF
\]
Substituting the expressions for \( ED \), \( DF \), and \( EF \):
\[
(6x - 4) + (3x + 5) = 46
\]
Simplifying the left-hand side:
\[
6x - 4 + 3x + 5 = 46
\]
\[
9x + 1 = 46
\]
Subtract 1 from both sides:
\[
9x = 45
\]
Now divide by 9:
\[
x = 5
\]

Now that we have \( x = 5 \), substitute it back into the expression for \( DF \):
\[
DF = 3x + 5 = 3(5) + 5 = 15 + 5 = 20
\]

So, \( DF = 20 \).

Thus, the correct answer is **A. 20**.

Would you like further details or explanations on any part of the solution? Here are five related questions to deepen your understanding:

1. How would the solution change if \( DF \) had a different expression, say \( DF = 4x + 3 \)?
2. What happens if \( EF \) were larger or smaller? How would that affect the value of \( x \)?
3. Can you explain the relationship between the segments in terms of a geometric diagram?
4. What if \( D \) were not between \( E \) and \( F \)? How would the approach change?
5. How would you solve if given three points in space and the problem asked for the 3D distance?

**Tip:** Always combine like terms carefully when dealing with algebraic expressions in word problems to avoid calculation errors.

Find DF if D is between E and F, ED = 6x - 4, DF = 3x + 5, and EF = 46.

This math problem involves finding DF using algebra and the segment addition postulate. Given ED = 6x - 4, DF = 3x + 5, and EF = 46, the solution involves solving for x and finding the length of DF. Suitable for Grades 7-9.

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