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Some true/false questions on absolute values

Decide whether the following are true or false.

  1. x < 5 \implies |x| < 5.
  2. |x-5| < 2 \implies  3 < x < 7.
  3. |1 + 3x| \leq 1 \implies x \geq -\frac{2}{3}.
  4. |x-1| = |x-2| is not satisfied by any x \in \mathbb{R}.
  5. For all x \in \mathbb{R}_{>0} there exists a y \in \mathbb{R}_{>0} such that |2x+y| = 5.

  1. False. Let x = -6, then x < 5, but |x| = |-6| = 6 > 5.
  2. True.

        \[ |x-5| <  2 \ \implies \ -(x-5) < 2 \qquad \text{and} (x-5) < 2 \]

    Hence,

        \[ x >3 \qquad \text{and} x < 7. \]

    This means 3 < x < 7.

  3. True.

        \[ |1+3x| \leq 1 \ \implies \ -\frac{2}{3} \leq x \leq 0 \ \implies \ -\frac{2}{3} \leq x. \]

  4. False. This equality is true for x = \frac{3}{2} since

        \[ \left| \frac{3}{2} - 1 \right| = \frac{1}{2} \qquad \text{and} \qquad \left| \frac{3}{2} - 2 \right| = \frac{1}{2}. \]

  5. False. Take x = 3, then

        \[ |2x+y| = 5 \ \implies \ |6+y| = 5 \ \implies \ y=-11 \text{ or } y = -1. \]

    Hence, y < 0 in either case and so y \notin \mathbb{R}_{>0}.

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