Describe the sets of real numbers geometrically

Describe the following sets of real numbers geometrically:

  • A = \{x : x < 7\}
  • B = \{x : |x| \ge 2\}
  • C = \{x : |x| = 1\}

Solution: Let A, B, C be as defined above.

  • The set A describes the ray extending from -\infty to 7, but not including 7.
  • Note that |x| \ge 2 \iff x \ge 2 or x \le -2. So B represents both a closed ray at -2 extending to \infty and a closed ray at 2 extending to \infty.
  • The set C describes the two points \pm 1 on the real line, because |x| = 1 \iff x = 1 or x=-1.
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