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Chris1122

Guest

Solving fraction inequality.

My reasoning here was that since the numerators are equal, then the two fractions will be equal when the denominators are equal and hence we obtain a solution x = -8. And now I reasoned that, again, since the numerators are equal, then the fraction on the left will be less than the fraction on the right when x-5 < 2x+3 from which it follows that x > -8 and so we obtain the final solution to the inequality that x >= -8.

Now my problem is that in all of my reasoning I used the fact that the numerators were the same, if this wasnt the case then none of my logic would had worked. How do I solve this kind of inequalities without using such logic as I used which is too dependent on this particular problem?

Thanks.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Solving fraction inequality.

And now I reasoned that, again, since the numerators are equal, then the fraction on the left will be less than the fraction on the right when x-5 < 2x+3

NO NO NO!! This only works if

and

are both positive or both negative. If one of

and

is positive and the other negative, your reasoning breaks down.

Examples:

(i)

and

This was your reasoning and it works because 2 and 3 are positive. So far so good.

(ii)

and

The denominators are both negative and your reasoning works again. Again, good.

(iii)

but

This time your reasoning fails because the denominators are of different signs!

is NOT the solution to your problem. Please try again.

Last edited by JaneFairfax (2009-08-27 22:17:14)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Solving fraction inequality.

The problem here is that when you multiply an inequality by a negative amount, you need to flip the sign.

eg. 2 < 3
    -2 > -3

What you've done is to multiply by (x-5) and by (2x-3), so to do it that way you'd need to split into three cases - when they're both positive, both negative and when they're one of each.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Solving fraction inequality.

A much easier way to solve these inequalities is to find all the places where they are either:

(i) Equal

OR

(ii) discontinuous

This will divide up the real line into intervals, and then you just test each one.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Chris1122

Guest

Re: Solving fraction inequality.

Is that the best way to do it? Is there an easier way?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Solving fraction inequality.

Hi chris1122;

These links show exactly how it is done:

http://www.sosmath.com/algebra/inequali … neq06.html
http://www.purplemath.com/modules/ineqrtnl.htm
http://www.wtamu.edu/academic/anns/mps/ … atineq.htm

Here is a good vid,watched it last night.
http://www.youtube.com/watch?v=SJecFvUbJOY

Do you understand why Jane is right. Yes, there is a very easy way to almost get this particular one but you should learn the book method first.

Last edited by bobbym (2009-08-28 01:09:27)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.