Solve |4x+12| > 16: Absolute Value Inequality Challenge

Given:

$\left|4x+12\right|>16$

Which of the following statements is necessarily true?

To solve this inequality, we use the principle that if $|A| > B$, then $A > B$ or $A < -B$. Here's a detailed step-by-step solution:

• Set up the two inequalities from the absolute value expression:
  1. $4x + 12 > 16$
  2. $4x + 12 < -16$
• For the first inequality, $4x + 12 > 16$:
  - Subtract 12 from both sides to isolate the term with $x$:
  $4x > 16 - 12$
  $4x > 4$
  - Divide both sides by 4:
  $x > 1$
• For the second inequality, $4x + 12 < -16$:
  - Subtract 12 from both sides:
  $4x < -16 - 12$
  $4x < -28$
  - Divide both sides by 4:
  $x < -7$
• The solution is the union of both results:
  $x > 1$ or $x < -7$

Therefore, the correct answer among the given choices is $x > 1$ or $x < -7$.