1/2^x < 0.01 Why is the sign flipped ?

Question

I have a problem and the solution book implies that the sign flips when simplification is performed. Why?:

1/2^x < 0.01 (1 over 2 to the power of x < 0.01)
-->
2^x > 1/0.01
-->
2^x > 100

Why did the flipping of the sign happen. To my understanding, the sign only flips when you divide or multiply by a negative number. Did I miss something here or is my book wrong ?

Thanks.

Answer 1

1/2^x < 0.01
then multiply both sides by 2^x
1 < 0.01 2^x
now divide both sides by 0,01:
1/ 0.01 < 2^x
100 < 2^x
~

Answer 2

Think about this:

1/2 > 1/4
2 < 4

5 > 1/2
1/5 < 2

8 > 3
1/8 < 1/3

Makes sense if you look at it with numbers. It has to do with them being reciprocals... which changes the order of the inequality.

If you think about a number line, notice that all numbers greater than 1 get "flipped" about the value 1 and end up between 0 and 1. If you have a really big x (e.g. 1 million) the reciprocal is really small (e.g. 0.000001). The bigger the x (e.g. 1 billion), the smaller the reciprocal (e.g. 0.000000001).

For two numbers, if 0 < a < b, then then reciprocals have the *opposite* relationship:
1/a > 1/b.

One thing to be careful of, though. If the smaller number is negative and the larger number is positive, the inequality would *not* flip. For example -2 < 2, but -1/2 < 1/2. In your case, this is not an issue. 2^x is positive and so is 0.01. If they are both negative, you can flip the sign. If they are both positive, you can flip the sign. If one is positive and the other is negative, don't flip it.

The last thing to obviously watch out for (same with equalities) is if one of the values could be zero, because the reciprocal would be undefined...

Answer 3

it is a property of inequations
if you have this : a < b (a*b>0) then 1/a > 1/b

1/2^x surely is larger than 0

Answer 4

The sign flips when you multiply/divide by ANY number, not just negative.