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Solve the following inequalities for x: 1) -3 <= x <= 2. 2) 2 - 3x >= x - 3.

Mathematics

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Here are the solutions for the given problems.

  1. Determine the value of xx satisfying −3≤x≤2-3 \le x \le 2.

This inequality already expresses the range of xx.

−3≤x≤2\boxed{-3 \le x \le 2}
  1. Determine the value of xx satisfying 2−3x≥x−32 - 3x \ge x - 3.

  1. Gather all terms involving xx on one side and constant terms on the other side.

    Subtract xx from both sides:

    2−3x−x≥x−3−x2 - 3x - x \ge x - 3 - x 2−4x≥−32 - 4x \ge -3

  2. Subtract 2 from both sides.

    2−4x−2≥−3−22 - 4x - 2 \ge -3 - 2 −4x≥−5-4x \ge -5

  3. Divide by -4.

    Remember to reverse the inequality sign when dividing by a negative number.

    −4x−4≤−5−4\frac{-4x}{-4} \le \frac{-5}{-4} x≤54x \le \frac{5}{4} x≤54\boxed{x \le \frac{5}{4}}
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