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Absolute value refers to the distance of a number from zero on the number line, regardless of direction. For any number \( a \), its absolute value is denoted as \( |a| \). It is always non-negative.
For example:
- \( |3| = 3 \) because 3 is 3 units away from 0.
- \( |-4| = 4 \) because -4 is 4 units away from 0.
The concept of absolute value helps us understand distances without considering the direction, which is crucial in solving inequalities involving absolute values.

Inequalities show the relationship between two expressions where one is greater than, less than, or possibly equal to the other. There are a few types of inequalities:
- \( a < b \) means a is less than b.
- \( a \leq b \) means a is less than or equal to b.
- \( a > b \) means a is greater than b.
- \( a \geq b \) means a is greater than or equal to b.
When dealing with absolute value inequalities like \( 0 \leq |b| \leq 6 \), we need to interpret the absolute value range. \( |b| \leq 6 \) means the distance from 0 to \( b \) cannot exceed 6 units. Hence, \( -6 \leq b \leq 6 \) includes all numbers within the distance of 6 units from 0.

A number line is a visual representation of numbers on a straight line. It helps us easily see the relationships between different numbers and their distances from each other.
When solving absolute value inequalities, a number line can be a handy tool. For our inequality \( 0 \leq |b| \leq 6 \), we can plot it on the number line.
First, mark the points \( -6 \) and \( 6 \). The solution includes all numbers between and including \( -6 \) and \( 6 \). This range is everything within the interval \( [-6, 6] \). Visualizing this helps us understand that any value \( b \) within this interval satisfies the inequality. \br> So, when we list values like \( -6, -3, 0, 3, 6 \), we are picking points within this specified range on the number line.