Inequalities

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Remember our facts about inequalities.

When finding the radius and open interval of convergence for Taylor series, it is important to be comfortable working with absolute values. This assignment reviews some of the basic properties.

I understand. I do not understand.

One of the most common mistakes that students make is that they treat the absolute value as a linear function. In practice, this leads to mistakes like the following:

Incorrect: If $|x - 2| < 3$ then $|x| < 5$ .

To see why this is incorrect, note that when $x = -1$ , we have $|x - 2| = 3$ .

Select the interval below that correctly shows all points $x$ such that $|x - 2| < 3$ .

Select the interval below that correctly shows all point $x$ such that $|x| < 5$ .

Write the inequality $|3x - 5| \le 1$ without using the absolute value sign.

$x \ge 4/3$ or $x \le 2$ $4/3 \le x \le 2$ $3x - 5 = \pm 1$ $x \le 4/3$ or $x \ge 2$

Write the inequality $|x + 2| > 6$ without using the absolute value sign.

$x + 2 > 6$ $x + 2 = \pm 6$ $x + 2 < -6$ or $x + 2 > 6$ $x > -8$ or $x < 4$

Solve the inequality $|2x + 5| \ge 10$ and write your answer in interval notation: $\answer {(-\infty , -15/2]} \cup [5/2, \infty )$

Solve the inequality $2 < |x - 2| < 5$ and write your answer in interval notation: $\answer {(-3, 0)} \cup \answer {(4, 7)}$

Find all values for $x$ that satisfy: $|x + 4| < 8$

$|x| < 4$ $0< x < 4$ $-4 < x < 4$ $-12 < x < 4$

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.

I understand I do not understand