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Basic Inequality Question - Multiplying Inequalities

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Basic Inequality Question - Multiplying Inequalities [#permalink] New post 26 Jun 2013, 07:29
Hi guys, I have a basic question that can probably be answered pretty quickly. I tried google searching but could not find my answer.

My question is, the guide has this rule: "Only multiply inequalities together if both sides of both inequalities are positive."

What if both inequalities were negative?

If m and n are both positive, is mn < 10?
(1) m < 2
(2) n < 5

For instance, in the example above what if both "m" and "n" were negative? Could you solve the problem? Why or Why not?
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Re: Basic Inequality Question - Multiplying Inequalities [#permalink] New post 26 Jun 2013, 08:13
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tmipanthers wrote:
Hi guys, I have a basic question that can probably be answered pretty quickly. I tried google searching but could not find my answer.

My question is, the guide has this rule: "Only multiply inequalities together if both sides of both inequalities are positive."

What if both inequalities were negative?

If m and n are both positive, is mn < 10?
(1) m < 2
(2) n < 5

For instance, in the example above what if both "m" and "n" were negative? Could you solve the problem? Why or Why not?


see this rule is because inequality sign changes when a negative thing is multipled...

example:
3 > 2===>now in this case if you multiple both sides with a negative number lets us suppose -2
then LHS=-6
RHS=-4
YOU CAN CLEARLY SEE THAT RHS > LHS ==>Initially LHS>RHS.
So guide is actually asking to keep caustion ...because we people forget to change the inequality some times.

now in your question:
If m and n are both negative, is mn < 10?
(1) m < 2
(2) n < 5

now as you if m and n are negative ..then there product will be postive..
product of two numbers <10
(1) m < 2==>we dont know about n hence ==not sufficient
(2) n < 5==> we dont know about M hence==not sufficient

now combining m is negative and m<2 and n is negative too and n<5
again not sufficient
take n=-10
m=-2===>mn=20==which is greater than 10.NO
Now take m=-1..n=-3
==mn=3===which is <10.YES.
NO DEFINITE ANSWER...HENCE NOT SUFFICIENT
ans E

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Re: Basic Inequality Question - Multiplying Inequalities [#permalink] New post 26 Jun 2013, 08:20
Thank you very much Shaileshmishra! That was very helpful. In fact, do you mind if I ask you another question on inequalities.

The guide also says: "Note that you can only take the square root of an inequality for which both sides are definitely not negative, since you cannot take the square root of a negative number."

I am confused about the scope of this rule.
If I am given this
"x^2>9"

Then that means |x| > 3

Can I not say then X > 3 or x <-3?
It seems I am taking the square root anyways but allowing for both a positive and negative possibilities of X. Is this correct? In what situation does the rule I just posted control if not this one?

I hope I made my question clear enough!
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Re: Basic Inequality Question - Multiplying Inequalities [#permalink] New post 26 Jun 2013, 08:36
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tmipanthers wrote:
Thank you very much Shaileshmishra! That was very helpful. In fact, do you mind if I ask you another question on inequalities.

The guide also says: "Note that you can only take the square root of an inequality for which both sides are definitely not negative, since you cannot take the square root of a negative number."

I am confused about the scope of this rule.
If I am given this
"x^2>9"

Then that means |x| > 3

Can I not say then X > 3 or x <-3?
It seems I am taking the square root anyways but allowing for both a positive and negative possibilities of X. Is this correct? In what situation does the rule I just posted control if not this one?

I hope I made my question clear enough!


hi,
see the book rule is perfectly fine.
actually the defenition of square root holds only for positive numbers, see the attached graph ==>it is defined only for positive values of X.
Now in your question
"x^2>9"....(1)
Then that means |x| > 3
this is perfectly done....
in equation one x^2==>this thing is greater than 9===>means we can say that it is positive===>x^2 is positive ..now as the rule says you can take the square root of positive number. so you have done perfectly fine.==>we dont have to worry about the value of x ===>we have to think as we are taking square root of x^2 then x^2 must be positive.
just keep always in your mind: sqrt of x^2 = mod of x==>as you have done.

one example :
x^2>-16....==> then in this case dont take sqr root of -16...as it is negative....this example lil weird ,but i think i made my point

kudos appreciated.
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GIVE VALUE TO OFFICIAL QUESTIONS...



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Re: Basic Inequality Question - Multiplying Inequalities [#permalink] New post 26 Jun 2013, 09:05
shaileshmishra wrote:
tmipanthers wrote:
Thank you very much Shaileshmishra! That was very helpful. In fact, do you mind if I ask you another question on inequalities.

The guide also says: "Note that you can only take the square root of an inequality for which both sides are definitely not negative, since you cannot take the square root of a negative number."

I am confused about the scope of this rule.
If I am given this
"x^2>9"

Then that means |x| > 3

Can I not say then X > 3 or x <-3?
It seems I am taking the square root anyways but allowing for both a positive and negative possibilities of X. Is this correct? In what situation does the rule I just posted control if not this one?

I hope I made my question clear enough!


hi,
see the book rule is perfectly fine.
actually the defenition of square root holds only for positive numbers, see the attached graph ==>it is defined only for positive values of X.
Now in your question
"x^2>9"....(1)
Then that means |x| > 3
this is perfectly done....
in equation one x^2==>this thing is greater than 9===>means we can say that it is positive===>x^2 is positive ..now as the rule says you can take the square root of positive number. so you have done perfectly fine.==>we dont have to worry about the value of x ===>we have to think as we are taking square root of x^2 then x^2 must be positive.
just keep always in your mind: sqrt of x^2 = mod of x==>as you have done.

one example :
x^2>-16....==> then in this case dont take sqr root of -16...as it is negative....this example lil weird ,but i think i made my point

kudos appreciated.

Superb explanation! Thank you again.
Re: Basic Inequality Question - Multiplying Inequalities   [#permalink] 26 Jun 2013, 09:05
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