Can someone explain this inequality transformation

Question

I have seen this transformation but am not sure what laws are applied to achieve it.

$$ \frac{1}{2^{n+1}} \leq 10^{-6} \Rightarrow2^{n+1} \geq 10^6. $$

I feel it is related to $x^{-1} = \frac{1}{x}$ and hence removing the negative exponents from both sides flips the inequality?

Multiply both sides by $2^{n+1} \cdot 10^6$. – VHarisop May 21 '18 at 18:28 — VHarisop, May 21 '18 at 18:28

score 2 · Accepted Answer · answered May 21 '18 at 18:30

2

Multiplying by $2^{n+1}>0$ and $10^6>0$ we get $$10^6\le 2^{n+1}$$

answered May 21 '18 at 18:30

Dr. Sonnhard Graubner

95,283

score 1 · Answer 2 · answered May 21 '18 at 18:31

1

Multiply by $2^{n+1}$ to get $$1\le10^{-6}\cdot2^{n+1}$$ Then divide by $10^{-6}$ to get $$10^6\le2^{n+1}$$ Note that the inequality will not flip because $2^{n+1}$ and $10^{-6}$ are both positive.

answered May 21 '18 at 18:31

Rhys Hughes

12,842

Can someone explain this inequality transformation

2 Answers2