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The problem asks us to evaluate the expression \( |5x + |2x + | -x ||| \) under the condition that \( x < 0 \).

### Step-by-step solution:

1. **Expression Breakdown:**
   - We have a nested absolute value expression. Let’s handle this layer by layer starting from the innermost absolute value.

2. **Innermost Absolute Value: \( |-x| \):**
   - Since \( x < 0 \), \( -x \) is positive. Therefore, \( |-x| = -x \).

3. **Next Level Absolute Value: \( |2x + |-x|| \):**
   - From the previous step, \( |-x| = -x \), so the expression inside this absolute value becomes:
     \[
     |2x + (-x)| = |x|.
     \]
   - Since \( x < 0 \), \( |x| = -x \).

4. **Outer Absolute Value: \( |5x + |2x + |-x||| \):**
   - Now substitute \( |2x + |-x| = -x \) into the outer expression:
     \[
     |5x + (-x)| = |4x|.
     \]
   - Again, since \( x < 0 \), \( |4x| = -4x \).

### Final Answer:
The value of the expression is \( -4x \).

Would you like further clarification or have any other questions? Here are some related questions for practice:

1. How do absolute values work when nested within other absolute values?
2. What happens if the condition was \( x > 0 \) in this problem?
3. Can you think of a general strategy for solving absolute value problems with variables?
4. How does the result of this problem change if the coefficients are different?
5. How would you represent this solution graphically on the number line?

**Tip:** Always simplify absolute value expressions by analyzing each layer carefully, starting from the innermost part.

x < 0 olduğuna göre, |5x + |2x + |-x||| ifadesinin değeri nedir?

Solve the nested absolute value expression |5x + |2x + |-x||| given that x < 0. This problem covers key algebraic concepts and theorems related to absolute value, with a detailed step-by-step breakdown. Suitable for Grades 9-11.

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