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Question 1208898: Prove that if 0 < a < b, then
0 < (1/b) < (1/a).
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52794)   (Show Source):
You can put this solution on YOUR website! .
Prove that if 0 < a < b, then
0 < (1/b) < (1/a).
~~~~~~~~~~~~~~~~~~~~~~~~
In the given inequality
0 < a < b
divide all three terms by positive number ab.
+-----------------------------------------------------------+
| Use well known basic property of inequalities |
| that if all sides of the given/valid inequality |
| are multiplied or divided by the same positive number, |
| then the updated inequality is also valid. *) |
+-----------------------------------------------------------+
You will get
0/(ab) < a/((ab) < b/(ab).
Reduce common factors, and you will get the desired inequality
0 < 1/b < 1/a.
Solved.
-------------------
*) This property of inequalities is a basic.
When you get an assignment like this one in the post,
it is assumed that you just know the basic necessary prerequisites.
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comment from student: You said: ". . .divide all three terms by positive number ab. "
How did you know what to do here?
What information in the question indicates to divide all three by positive ab?
My response: You ask " How did you know what to do here? "
For my solution and for your learning, it does not matter how I do know it.
Probably, I learned it from my Math teacher/teachers; or from my school Math textbook/textbooks; or from many
other mathematical books that I read; or, may be, I was smart enouigh to invent this trick on my own.
What   important - is the fact that it is  way solving.
It is way solving. It is way solving.
Now, as you read it from my post, you also know this way thinking.
It is how the process of spreading knowledge works:
- I learn something from some source;
- I know that it is right - because it is my expert's suggestion;
- I share and disseminate my knowledge;
- after reading my post, you know it.
So, your question "how do I know it?" is good, but it sounds in dissonance to my post.
It is the same as after explanation in the class, given by your teacher,
you ask him or her: "how do you know it?" - In response, the teacher will look at you
and shrug his shoulders. - - - Simply because the teacher spent years of his life
to absorb knowledge and to learn the subject.
In order for the communication be harmonic, you should say "Thanks" at the end.
Then you will reach a harmony.
In general, many of your comments remind me of an orchestra, where, when
the flutes are supposed to play, suddenly the sound of a drum is heard, instead.
Answer by MathTherapy(10552)  (Show Source):
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