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.