SOLUTION: I need some help solving this inequality. {{{x-5/3-x}}}<0 Thank you so much!

Question 643496: I need some help solving this inequality.
<0
Thank you so much!
Found 2 solutions by MathTherapy, KMST:
Answer by MathTherapy(10555) (Show Source): You can put this solution on YOUR website!
I need some help solving this inequality.
<0
Thank you so much!

Before trying to solve this inequality, upon observing it, we realize that whatever x-value one uses, that x-value becomes 0. This is because we are adding x, and also subtracting it (see inequality). This leaves us with: , which is always, and will forever be TRUE. This means that ANY x-value inserted into the inequality will satisfy it (make it true).

Anyway, still working the problem out, we convert the left-side expression to one with a common denominator, 3 to get:

----- 3x - 3x in numerator equals 0

Since this is always true, ANY value that is plugged in for x will satisfy the inequality, thereby leading to the solution x = ALL REALS.

Send comments and �thank-yous� to �D� at MathMadEzy@aol.com
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Typo, or trick question?

IF THE PROBLEM WAS REALLY
x-5/3-x<0 , which is the same as ,
Then those and cancel each other.
You are left with ,
which is true for any value of
So if that was the problem, the answer is x = any real number.

I SUSPECT THE THE PROBLEM MAY HAVE BEEN
, which is something different,
and can be written as (x-5)/(3-x)<0 .
In that case, the problem is not so simple.
Then, we have to think of different cases, and look at when the numerator and denomiantor are positive, negative or zero.

I consider the cases (x<3, x=3, x between 3 and 5, x-5, and x>5).
For each case, I figure out if , the numerator of the rational expression , is positive, negative, or zero.
I figure out if , the denominator, is positive, negative, or zero.
From there, I figure out if the whole rational expression will be positive, negative, zero, or undefined. If numerator and denominator have the same sign (both positive, or both negative), dividing one by the other, I get a positive value for the rational expression . If numerator and denominator have opposite signs, then the rational expression will be negative.
To keep my thinking straight, I make myself a table like this:

MORE EXPLANATIONS:
(Feel free to skip them; I usually explain a lot more than needed, and bore people to tears).

LOOKING AT THE DENOMINATOR:
If <---> , then the rational expression is undefined, because it has zero for a denominator.
For , then and for , then .

LOOKING AT THE NUMERATOR:
For , then .
For , then .
For , then .

FOR THE WHOLE RATIONAL EXPRESSION:
Dividing (or multiplying) two expressions with opposite signs (one positive and one the other negative number), the result is negative,
If both expressions have the same sign (both positive, or both negative, their product or quotient is positive.

THE CASES:
We have to see what happens when is less than 3, at 3, between 3 and 5, at 5, and when is more than 5.
For , , while , and .
In that case , telling us that is part of the solution
For , the expression does not exist.
For between 3 and 5, nummerator and denominator are both negative,
so .
For , and
For , , while , so , so is negative again.
In that case (), telling us that is part of the solution.
The solution is or .
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