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Absolute Value
The concept of absolute value is best understood as that of distance.
Part 1. On the familiar number line, such as the one below,
the absolute value of x is defined to be the distance between x and the origin. Since the distance can never be negative, the absolute value of a number is always either 0 or positive. For example,
| –5 | = Distance
between –5 and the origin =
5 units.
Part 2. Being analogous to Part 1, we define | x – a | to be the distance between x and a on the number line. Part 1 is a special case when a = 0.
Be careful with absolute values such as | x + 7 | because we need to interpret it as
| x – (–7) | with a = –7. Namely,
| x +
7 | = Distance between x and –7 .
Part 3. We now consider the absolute-vaule equation | x + 7 | = 10. Relying on what was explained in Part 2, we see that x must be 10 units away from –7 either to the left or to the right. So, x = –17 or x = 3. That is why when the equation is solved algebraically, the solution will look like this:
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Disclaimer • Updated September 4, 2008
Email: mitsuma@kutztown.edu
Phone: +1 (610) 462-WPJW
Phone: +1 (610) 462-WPJW