Prove that: \( |a| \geq 9 \)

To prove that \( |a| \geq 9 \), we can start by analyzing the definition of absolute value. The absolute value \( |a| \) represents the distance of \( a \) from zero on the number line. Therefore, if \( |a| \geq 9 \), it means that \( a \) is either greater than or equal to 9 or less than or equal to -9. We can express this with the inequalities: 1. \( a \geq 9 \) or 2. \( a \leq -9 \) Thus, unless \( a \) falls within the range \(-9 < a < 9\), which contradicts the absolute value condition, we can confidently conclude that \( |a| \) must indeed be greater than or equal to 9. One way to demonstrate this is by considering specific cases of values for \( a \) that satisfy the conditions of the absolute value, confirming that they will produce values meeting or exceeding 9.