If DE = 2, EF = 2x, and DF = x + 3, what is DF?

Step-by-step solution

1. 1

   Set up with the Segment Addition Postulate. With $E$ between $D$ and $F$, the two pieces $DE$ and $EF$ make up the whole $DF$:

   $DE+EF=DF$

   What makes this version slightly harder than usual is that $x$ appears on both sides — the whole segment is unknown too.

2. 2

   Substitute and gather the variable.

   $2+2x=x+3$

   Subtract $x$ from both sides:

   $2+x=3\;\Longrightarrow\;x=1$

3. 3

   Compute the requested length. The question asks for $DF$, so substitute $x=1$ into the expression for $DF$:

   $DF=x+3=1+3=4$

4. 4

   Verify by measuring both pieces. With $x=1$ the parts are $DE=2$ and $EF=2(1)=2$, and

   $2+2=4=DF\;\checkmark$

   The point $E$ turns out to be the midpoint of $\overline{DF}$, since the two pieces come out equal — a fact the algebra reveals but the problem statement never announces.