Solve ||1-4|+3|-|a|<0: Complex Absolute Value Inequality

To solve this problem, we'll follow these steps:

• Step 1: Simplify the expression inside the absolute values.
• Step 2: Analyze the inequality and interpret the result.
• Step 3: Determine which condition on $a$ satisfies the inequality.

Now, let's work through each step:

Step 1: Simplify the expression inside the absolute values.
Inside the first absolute value, calculate $|1-4|$. We have $1-4 = -3$, so $|-3| = 3$.
Now, calculate $|3+3| = |6| = 6$.
Thus, the expression becomes |6 - |a|| < 0 .

Step 2: Analyze the inequality.
The absolute value of any real number is non-negative, meaning $|6 - |a|| \geq 0$.
The inequality |6 - |a|| < 0 suggests that it's impossible to have a non-negative number less than 0 unless it results in exactly zero, which isn’t possible here.
However, for this particular structure, note if $6 - |a| \neq 0$, the inequality comes from where an incorrect assumption in formulation.

Step 3: Solving the inequality.
For 6 - |a| < 0 , we solve for $a$:
6 < |a|

This inequality $|a| > 6$ means:

• $a > 6$
• or $a < -6$

Therefore, the solution to the problem is that $a$ must satisfy $a > 6$ or $a < -6$.

Therefore, the correct choice is: $a > 6$ or $a < -6$.