Simplifying Absolute Values: Which Statement Holds True?

Given:

$|a|+||5-1|+3-4|<0$

Which of the following statements is necessarily true?

To solve this problem, we start by analyzing the inequality:

• Simplify the expression inside absolute values:
  $|5-1| = |4| = 4$
  $|3-4| = |-1| = 1$
• Evaluate the inner expression:
  $||5-1|+3-4| = |4 + 1| = |5| = 5$
• Substitute back into the full expression:
  $|a|+5 < 0$

According to the properties of absolute values, $|a|$ is always non-negative, so it can only add to 5 or keep it positive.

Therefore, the only value this expression can assume is non-negative. Hence, it can never be less than zero.

Consequently, the original condition $|a|+5 < 0$ is impossible.

The correct answer is that the inequality has no solution.

No solution