The image shows a geometric line segment with points labeled $S$, $T$, and $U$. The distances are marked as follows:

• From $S$ to $T$, the distance is given as 8.
• From $T$ to $U$, the distance is represented as $2x + 19$.
• The total distance from $S$ to $U$ is $x + 19$.

To solve for $x$, we can use the fact that the sum of the parts of the segment must equal the total length:

$ST + TU = SU$

Substituting the given values:

$8 + (2x + 19) = x + 19$

Now, let's solve for $x$:

1. Combine like terms on the left-hand side: $8 + 2x + 19 = x + 19$ $2x + 27 = x + 19$

2. Subtract $x$ from both sides: $2x - x + 27 = 19$ $x + 27 = 19$

3. Subtract 27 from both sides: $x = 19 - 27$ $x = -8$

So, $x = -8$.

Would you like more details or explanations? Here are five related questions to explore:

1. How can you check the solution by plugging $x = -8$ back into the equation?
2. What does a negative value for $x$ represent in this problem?
3. How does this problem illustrate the Segment Addition Postulate?
4. Could you solve this equation using a different algebraic method?
5. How would the problem change if $SU$ were a different expression?

Tip: Always double-check your solution by substituting it back into the original equation to ensure it satisfies the conditions.