Given that $E$ is between $D$ and $F$, we can deduce that the length of segment $DF$ is the sum of the lengths of segments $DE$ and $EF$.

Information provided:

• $DE = x + 1$
• $EF = x + 7$
• $DF = 4x - 20$

Since $DF = DE + EF$, we can set up the equation: $DE + EF = DF$ Substitute the given values: $(x + 1) + (x + 7) = 4x - 20$

Simplify the left-hand side: $2x + 8 = 4x - 20$

Now, solve for $x$:

1. Subtract $2x$ from both sides: $8 = 2x - 20$
2. Add 20 to both sides: $28 = 2x$
3. Divide both sides by 2: $x = 14$

Final Answer:

$x = 14$.

Would you like a detailed breakdown of each step?

Here are 5 related questions to deepen understanding:

1. What is the length of $DE$ when $x = 14$?
2. What is the length of $EF$ when $x = 14$?
3. What is the total length of $DF$ when $x = 14$?
4. How can you verify if the segment addition is correct for this problem?
5. What happens if $E$ is not exactly between $D$ and $F$?

Tip: Always check if the segment addition formula applies when points are collinear.