To solve for $DE$, we need to use the fact that $D$, $E$, and $F$ are on the same line, so the sum of the lengths $DE$ and $EF$ must equal the total length $DF$.

We are given:

• $DE = x - 4$
• $EF = 12$
• $DF = 2x - 11$

From this, we know: $DE + EF = DF$ Substituting the known values into the equation: $(x - 4) + 12 = 2x - 11$

Now, let's solve for $x$:

$x - 4 + 12 = 2x - 11$ $x + 8 = 2x - 11$ Subtract $x$ from both sides: $8 = x - 11$ Add 11 to both sides: $x = 19$

Now that we know $x = 19$, substitute this back into the expression for $DE$:

$DE = x - 4 = 19 - 4 = 15$

Thus, $DE = 15$.

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:

1. What is the length of $DF$ when $x = 19$?
2. How would the solution change if $EF$ was doubled?
3. What if $DF$ was expressed as $3x - 15$, how would the equation adjust?
4. If $EF = x$, how would that affect the final solution for $DE$?
5. Can you verify the solution using a different method, like graphing?

Tip: When dealing with line segments, always check if points are collinear to ensure that the segment lengths add up properly.