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Mathematics

Which one of the following statements is incorrect?

  1. If a < b, then a - m < b - m

  2. If a > b and m > 0, then am > bm

  3. If a < b and m > 0, then am\dfrac{a}{m} > bm\dfrac{b}{m}.

  4. If a ≠ 0 and b ≠ 0, then a > b ⇒ 1a\dfrac{1}{a} < 1b\dfrac{1}{b}.

Linear Inequations

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Answer

Let's evaluate each:

  1. If a < b, then a - m < b - m
    Adding/Subtracting doesn't change the sign. So, it is correct.

  2. If a > b and m > 0, then am > bm
    Multiplying by a positive number keeps the sign. So, it is correct.

  3. If a < b and m > 0, then am\dfrac{a}{m} > bm\dfrac{b}{m}.
    Incorrect. If we divide by a positive number, the sign should remain the same. It should be am\dfrac{a}{m} < bm\dfrac{b}{m}.

  4. If a ≠ 0 and b ≠ 0, then a > b ⇒ 1a\dfrac{1}{a} < 1b\dfrac{1}{b}.
    Generally correct for positive numbers (Reciprocal rule).

Option 3 is the incorrect statement.

Hence, option 3 is the correct option.

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