Question 2.31: Describe the interval on the x axis defined by (a) |x| <......

(a)  |x| < 2 is the same statement as —2 < x < 2; that is, x is numerically less than 2.
Figure 2.39 illustrates this region. Note that the region is an open interval. Since the points x = —2 and x = 2 are not included, they are shown on the graph as o.

(b) If |xl ≥ 3 then either x ^ 3 or x ≤ — 3. This is shown in Figure 2.40. The required region of the x axis has two distinct parts. Since the points x = 3 and x = —3 are included in the interval of interest, they are shown on the graph as •.

(c) |x — 1| < 3 is equivalent to —3 < x — 1 <3, that is —2 < x <4.

(d) |x + 2| > 1 is equivalent to x + 2 > 1 or x + 2 < —1, that is x > —1 or x < —3.