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# 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 $x=-1$.