The problem presented is an absolute value problem under the condition $x < 0$. The expression given is:

$\left| 5x + \left| 2x + \left| -x \right| \right| \right|$

Step-by-Step Solution:

1. Evaluate $|-x|$: Since $x < 0$, $-x$ becomes positive. Therefore: $| -x | = -x$

2. Substitute $|-x|$ into the middle absolute value: Now, substitute $| -x | = -x$ into the second part of the expression: $\left| 2x + | -x | \right| = \left| 2x - x \right| = | x |$ Again, since $x < 0$, the absolute value of $x$ is $-x$: $| x | = -x$

3. Substitute into the outermost absolute value: Now substitute $| x | = -x$ into the outermost part of the expression: $\left| 5x + | 2x + | -x | | \right| = \left| 5x + (-x) \right| = | 4x |$ Since $x < 0$, we know that: $| 4x | = -4x$

Final Answer:

The value of the expression is $-4x$.

Do you have any questions or would you like more details?

Here are 5 related questions for further practice:

1. What is the value of $\left| 3x + \left| x + 2 \right| \right|$ when $x < 0$?
2. How would the expression change if $x > 0$?
3. Can you solve the expression $\left| -3x + 2 \right|$ when $x < 0$?
4. What is the derivative of $|x|$ with respect to $x$?
5. Solve the expression $|2x + 5| = 10$ for $x$.

Tip:

When dealing with absolute value expressions, always consider the sign of the variable inside to simplify the calculations.