In the given problem, we are provided with three segment lengths:

• $KL = 7x$
• $LM = 4x - 4$
• $KM = 6x + 16$

Since $K$, $L$, and $M$ are points on a straight line, the sum of $KL$ and $LM$ must equal $KM$. Hence, we can set up the equation:

$KL + LM = KM$

Substitute the expressions for $KL$, $LM$, and $KM$:

$7x + (4x - 4) = 6x + 16$

Now, solve for $x$:

$7x + 4x - 4 = 6x + 16$

$11x - 4 = 6x + 16$

Subtract $6x$ from both sides:

$5x - 4 = 16$

Add 4 to both sides:

$5x = 20$

Now, divide by 5:

$x = 4$

Now that we have $x = 4$, substitute this value back into the expression for $LM$:

$LM = 4x - 4 = 4(4) - 4 = 16 - 4 = 12$

Thus, the length of $LM$ is 12 units.

Would you like further details or explanations?

Here are some related questions:

1. What would happen if $KM$ were shorter than the sum of $KL$ and $LM$?
2. How do we know that $K$, $L$, and $M$ are collinear?
3. If $x = 3$, what would the value of $LM$ be?
4. Can you express $KL$ and $KM$ in terms of $LM$ alone?
5. What is the geometric significance of solving this type of equation?

Tip: When solving equations, combining like terms as early as possible simplifies the process of finding the solution.