https://file.aiquickdraw.com/mathbot/file/1724113184659h6g8dj0e.jpg

Let's solve the problem step by step.

### Problem Analysis
The diagram shows a line segment \(NXZ\) with points \(M\) and \(L\) lying on it. The lengths of the segments are given as follows:

- \(NM = 110 + x\)
- \(ML = x + 113\)
- \(LZ = 10\)
- \(NZ = x + 253\)

We need to find the length of segment \(ML\).

### Steps to Solve
1. **Expression for \(NZ\)**: 
   The total length \(NZ\) can be expressed as the sum of the lengths of the segments \(NM\), \(ML\), and \(LZ\):
   \[
   NZ = NM + ML + LZ
   \]
   Substitute the given expressions:
   \[
   x + 253 = (110 + x) + (x + 113) + 10
   \]

2. **Simplify the Equation**:
   Combine the terms involving \(x\) and the constant terms:
   \[
   x + 253 = x + 110 + x + 113 + 10
   \]
   \[
   x + 253 = 2x + 233
   \]

3. **Solve for \(x\)**:
   To isolate \(x\), subtract \(x\) from both sides:
   \[
   253 = x + 233
   \]
   Subtract 233 from both sides:
   \[
   x = 20
   \]

4. **Find \(ML\)**:
   Now that we know \(x = 20\), substitute it back into the expression for \(ML\):
   \[
   ML = x + 113 = 20 + 113 = 133
   \]

### Final Answer
The length of segment \(ML\) is \(133\) units.

Would you like any further details or have any questions?

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Here are some related questions to consider:

1. What would the length of \(NZ\) be if \(x\) were 25?
2. How would you solve for \(x\) if the total length \(NZ\) was unknown?
3. Could you use a different method to verify the length of \(ML\)?
4. What would happen to the solution if \(LZ\) was a different length?
5. How would the problem change if point \(Z\) was moved to the left of \(L\)?

**Tip:** When dealing with algebraic expressions, always make sure to simplify correctly to avoid calculation errors.

Learn how to find the length of segment ML on line segment NXZ with a detailed step-by-step solution. This problem involves algebraic expressions for segment lengths and is suitable for advanced high school students.

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