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How does absolute value mean distance? Why does |x – a| mean the distance of x from a?

Let's first discuss the term absolute value. The absolute value is fundamentally about the distance between two values, depending on the expression contained within the absolute value sign. For example, the distance between two numbers x and y can be written as |x – y|. Therefore:

  • |x| = |x – 0| is the distance of x from 0, the origin.
  • |x – 5| is in fact the distance of (x – 5) from the origin. So another way of interpreting this is that |x – 5| is the distance between x and 5

Let's look at some more examples:

  • |3 – 5| = |-2| = 2. In other words, 3 and 5 are a distance of 2 units apart.
  • |-2 – 6| = |-8| = 8. In other words, -2 and 6 are a distance of 8 units apart.
  • |4 – (-7)| = |4 + 7| = |11|. In other words, 4 and -7 are a distance of 11 units apart.

To summarize, the distance between two numbers x and y is defined by a subtractive relationship: |x – y|. Thus, when our 'y' term is positive (like 5), the distance between x and 5 is simply |x – 5|. When our 'y' term is negative (like -5), the subtraction sign gets converted to a plus sign, and the distance between x and -5 can be written as: |x – (-5)| = |x + 5|.

For further reading on absolute values, consider the following:

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