Inequalities

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The expression $|x - c| < r$ can be thought of in a more geometric way as well; it represents the collection of all points that are a distance $r$ units from a central point $c$ .

For example, if we consider the number line corresponding to the expression $|x - 4| < 2$

The center of this interval is at $x = 4$ . The endpoints $x = 2$ and $x = 6$ are precisely a distance of 2 aways from the center $x = 4$ . All points that satisfy the inequality are within a distance of 2 from the center.

This observation is particularly helpful when considering the interval of convergence of a Taylor series.

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