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TMUA 2019 · Paper 2 · Question 8 of 20

TMUA 2019 Paper 2 Question 8

Inequalities and reasoning — Multiplying and dividing by negatives. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 2Inequalities and reasoningMultiplying and dividing by negatives8 options
a, b and c are real numbers with a < b < c < 0.

Which of the following statements must be true?

I    ac < ab < a2
II   b(c + a) > 0
III  cb > ab

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · E
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
All three numbers are negative, so every multiplication or division by one of them flips the inequality. For I, multiply a < b < c through by the negative number a: the order reverses to a2 > ab > ac, which is exactly statement I. For II, b < 0 and c + a < 0, and a negative times a negative is positive, so II holds. For III, divide a < c by the negative number b: the order reverses to ab > cb, which is the opposite of what III claims. So I and II only.