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13 Feb 2023, 01:17
Kudos
Bookmarks
Hi Karishma 🙏
please advise in x-1/x<0
(X^2-1)/x<0
(x-1)(x+1)/x<0
Now x cannot be zero. As denominator can not zero
But how do we get
0 < x < 1 or x < -1
I understand that x must be positive
Posted from my mobile device[/quote][/quote]
Check out this post:
https://gmatclub.com/forum/inequalities ... 91482.html
Let me know if you have any doubts in it.[/quote]
KarishmaB
Hi Karishma
So this is what I got from the link you have shared with me.
From (x-1)(x+1)/x < 0
We get three points on number line ---(-1)-----0------1----
So we four ranges x>1 0<x<1 -1<x<0 and x<-1
Now if x is greater than 1 equation will be positive hence greater then
zero---> not possible
If x is between zero and 1
(x-1) is negative and x+1
positive hence equation is negative and less then 1
Say x =.5 then it becomes
(-0.5)(1.5)/0.5= -1.5
If x is between -1 to 0
Then equation becomes postive
Say x is -0.5
Then -(1.5)(0.5)/-0.5= positive hence greater then greater then 0 again not possible also x cannot be less then zero.
If x is less then -1 then equation is still negative it will (-ve)(-ve)/-ve still negative I've and less then 0
But x cannot be negative hence x between 0-1
please advise in x-1/x<0
(X^2-1)/x<0
(x-1)(x+1)/x<0
Now x cannot be zero. As denominator can not zero
But how do we get
0 < x < 1 or x < -1
I understand that x must be positive
Posted from my mobile device[/quote][/quote]
Check out this post:
https://gmatclub.com/forum/inequalities ... 91482.html
Let me know if you have any doubts in it.[/quote]
KarishmaB
Hi Karishma
So this is what I got from the link you have shared with me.
From (x-1)(x+1)/x < 0
We get three points on number line ---(-1)-----0------1----
So we four ranges x>1 0<x<1 -1<x<0 and x<-1
Now if x is greater than 1 equation will be positive hence greater then
zero---> not possible
If x is between zero and 1
(x-1) is negative and x+1
positive hence equation is negative and less then 1
Say x =.5 then it becomes
(-0.5)(1.5)/0.5= -1.5
If x is between -1 to 0
Then equation becomes postive
Say x is -0.5
Then -(1.5)(0.5)/-0.5= positive hence greater then greater then 0 again not possible also x cannot be less then zero.
If x is less then -1 then equation is still negative it will (-ve)(-ve)/-ve still negative I've and less then 0
But x cannot be negative hence x between 0-1
13 Feb 2023, 01:32
Kudos
Bookmarks
puneetfitness
Yes, here is an excerpt from my Inequalities book:
How to Handle an Inequality with Multiple Factors
Consider any inequality of the form (x - a)(x - b)(x - c) < 0. For clarity, let’s work with the example x(x - 3) < 0 discussed above.
Step 1: Make a number line and plot the points a, b and c on it. In our example, a = 0 and b = 3. The number line is divided into sections by these points. In our example, it is divided into 3 sections – ‘greater than 3’, ‘between 0 and 3’ and ‘less than 0.’
Step 2: Starting from the rightmost section, mark the sections with alternate positive and negative signs. The rightmost section will be positive, the next one will be negative, the next one will be positive and so on... The inequality will be positive in the sections where we have the positive signs and it will be negative in the sections where we have the negative signs.
Attachment:
Screenshot 2023-02-13 at 1.59.50 PM.png [ 16.58 KiB | Viewed 1531 times ]
Therefore, x*(x - 3) will be positive in the ranges ‘x > 3’ and in ‘x < 0’.
It will be negative in the range ‘0 < x < 3’.
Hence, the values of x for which x*(x - 3) < 0 is satisfied is 0 < x < 3.
Explanation:
When we plot the points on the line, the number line is divided into various sections. Values of x in the right most section will always give us positive value of the expression. The reason for this is that if x > 3, all factors will be positive i.e. x and (x - 3), both will be positive.
When we jump to the next region i.e. between x = 0 and x = 3, the values of x will give us negative values for the entire expression because now, only one factor, (x - 3), will be negative. All other factors will be positive.
When we jump to the next region on the left where x < 0, expression will be positive again because now both factors x and (x - 3) are negative. The product of two negatives is positive so the expression will be positive.
Similarly, we can solve a question with any number of factors.
13 Feb 2023, 01:41
Kudos
Bookmarks
KarishmaB
karishma great that was very helpful one last question what if inequality was
x(x - 3) > 0 instead of
x(x - 3) < 0
Will it still hold true
