Math Problem Statement

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

Solution

The problem asks us to evaluate the expression 5x+2x+x|5x + |2x + | -x ||| under the condition that x<0x < 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|-x|:

    • Since x<0x < 0, x-x is positive. Therefore, x=x|-x| = -x.
  3. Next Level Absolute Value: 2x+x|2x + |-x||:

    • From the previous step, x=x|-x| = -x, so the expression inside this absolute value becomes: 2x+(x)=x.|2x + (-x)| = |x|.
    • Since x<0x < 0, x=x|x| = -x.
  4. Outer Absolute Value: 5x+2x+x|5x + |2x + |-x|||:

    • Now substitute 2x+x=x|2x + |-x| = -x into the outer expression: 5x+(x)=4x.|5x + (-x)| = |4x|.
    • Again, since x<0x < 0, 4x=4x|4x| = -4x.

Final Answer:

The value of the expression is 4x-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>0x > 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.

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Math Problem Analysis

Mathematical Concepts

Absolute Value
Algebraic Expressions

Formulas

Absolute Value Property: |a| = a if a >= 0, |a| = -a if a < 0

Theorems

Absolute Value Theorem

Suitable Grade Level

Grades 9-11