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Unformatted text preview: or c > 0, |x| < c is equivalent to −c < x < c.
• For c ≤ 0, |x| < c has no solution.
• For c ≥ 0, |x| > c is equivalent to x < −c or x > c.
• For c < 0, |x| > c is true for all real numbers.
As with Theorem 2.1 in Section 2.2, we could argue Theorem 2.3 using cases. However, in light
of what we have developed in this section, we can understand these statements graphically. For
instance, if c > 0, the graph of y = c is a horizontal line which lies above the x-axis through (0, c).
To solve |x| < c, we are looking for the x values where the graph of y = |x| is below the graph
of y = c. We know the graphs intersect when |x| = c, which, from Section 2.2, we know happens
when x = c or x = −c. Graphing, we get
(−c, c) −c (c, c) c x We see the graph of y = |x| is below y = c for x between −c and c, and hence we get |x| < c is
equivalent to −c < x < c. The other properties in Theorem 2.3 can be shown similarly.
Example 2.4.3. Solve the fo...
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