If LM = 6, MN = x + 9, and LN = 6x, what is MN?

Step-by-step solution

1. 1

   Apply the postulate. Since $M$ is between $L$ and $N$,

   $LM+MN=LN$

   and substituting the three expressions turns a geometry statement into a one-variable equation:

   $6+(x+9)=6x$

2. 2

   Combine like terms.

   $x+15=6x$

3. 3

   Solve. Subtract $x$ from both sides and divide by $5$:

   $15=5x\;\Longrightarrow\;x=3$

4. 4

   Evaluate the requested segment.

   $MN=x+9=3+9=12$

5. 5

   Check every length. $LM=6$, $MN=12$, $LN=6(3)=18$, and $6+12=18\;\checkmark$. As a sanity test on the setup, note that $MN$ came out twice $LM$; if the solved $x$ had made any length negative, or a part longer than the whole, the labelling would have been inconsistent and the problem unsolvable as posed.