If LM = 9, MN = x + 7, and LN = 3x, what is MN?

Step-by-step solution

1. 1

   Identify the whole and the parts. The three names $LM$, $MN$, $LN$ tell you $M$ is between $L$ and $N$, so the Segment Addition Postulate reads

   $LM+MN=LN$

2. 2

   Substitute and simplify.

   $9+(x+7)=3x\;\Longrightarrow\;x+16=3x$

   The two constants $9$ and $7$ combine into $16$; keeping them separate is a common source of slips.

3. 3

   Solve for $x$.

   $16=2x\;\Longrightarrow\;x=8$

4. 4

   Substitute back to get $MN$.

   $MN=x+7=8+7=15$

   Note that $x=8$ is not a length here — it is only a parameter, and the length $MN$ happens to be nearly twice it.

5. 5

   Confirm the whole equals the sum of the parts. $LM=9$, $MN=15$, $LN=3(8)=24$, and $9+15=24\;\checkmark$. All lengths are positive and $LN$ is the largest, exactly as the betweenness of $M$ requires.