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TMUA 2016 · Paper 2 · Question 10 of 20

TMUA 2016 Paper 2 Question 10

Inequalities and reasoning — Sufficient conditions · which powers preserve order. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Inequalities and reasoningSufficient conditions · which powers preserve order6 options
In this question x and y are non-zero real numbers.

Which one of the following is sufficient to conclude that x < y?

  1. Ax4 < y4
  2. By4 < x4
  3. Cx−1 < y−1
  4. Dy−1 < x−1
  5. Ex35 < y35
  6. Fy35 < x35
Show the answer and worked solution
answer · E
  1. Ax4 < y4
  2. By4 < x4
  3. Cx−1 < y−1
  4. Dy−1 < x−1
  5. Ex35 < y35
  6. Fy35 < x35
The question is which map t ↦ tk is strictly increasing on the whole of the non-zero reals, since only then can the inequality be undone. Fourth powers destroy sign information: x = 1, y = −2 satisfies x4 < y4 but not x<y, and B fails similarly. Reciprocals reverse order on each side of zero but not across it: y−1 < x−1 holds for x = 1, y = −1, where x > y, and x−1 < y−1 holds for x = 2, y = 1. But t35 = (t15)3 is defined and strictly increasing for every real t, so x35 < y35 does force x < y.