We are given the following equations based on the line segment $SU$ with point $T$ on it:

• $ST = 4x - 4$
• $TU = 4x - 10$
• $SU = 5x + 1$

Since $T$ is on the line segment $SU$, we know that the total length of $SU$ must be the sum of the lengths of $ST$ and $TU$. Therefore:

$SU = ST + TU$

Substitute the given expressions for $ST$, $TU$, and $SU$:

$5x + 1 = (4x - 4) + (4x - 10)$

Now, solve this equation for $x$.

Step 1: Simplify the right-hand side

$5x + 1 = 4x - 4 + 4x - 10$ $5x + 1 = 8x - 14$

Step 2: Rearrange the equation to isolate $x$

$5x + 1 = 8x - 14$ $1 + 14 = 8x - 5x$ $15 = 3x$ $x = 5$

Step 3: Calculate $TU$

Now that we know $x = 5$, substitute this value into the expression for $TU$:

$TU = 4x - 10 = 4(5) - 10 = 20 - 10 = 10$

Thus, the numerical length of $TU$ is 10.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How do you verify if a point lies on a line segment?
2. Can line segments have negative lengths in geometric problems?
3. How can we interpret the relationship between variables and geometric distances?
4. If $ST$ and $TU$ had different forms, how would we adjust our approach?
5. How can we generalize this problem for other types of segments?

Tip: Always double-check your solution by substituting the value of $x$ back into the original conditions to confirm consistency.