Hi dear friends!
How are you?
Are you interested in "Math"?!
So…
Let's study it together!
Please answer these questions:
1- If we have:
1< (2|x|-1)/|x|+1 < 2
Then;
Which one is right?
1) x > 1/2
2) -2 < x < 2
3) x < -(1/3) or x > (1/3)
4) x > 2 or x < -2
******* ******* ******* ******* *******
2- If we have:
1< 4/(3x-2) < 2
Then;
Which interval is right about x?
1- (5/6,1[
2- [5/6,1)
3- (5/6, 1)
4- [5/6,1[
Replies
We can inverse the fraction. Also since 1 and 2 are positive numbers , we should change < into >. [1<2 so 1>1/2]
Now we have : 1 > (3x-2)/4 > 1/2
Multiplication of the condition above by 4 gives: 4> 3x-2 > 2 then 6 > 3x > 4 then 4/3<x<2!
I think the choices are wrong since x must be larger than 8/6 which says that 5/6 and above it which are less than 8/6 can't be the answer.
Am I right ?
I solved it another way, too. The same answer was resulted!
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Yes!!
You are right exactly!
Excellent!
Thanks dear.
:-*
First of all, I am really happy to have such an intelligent girl here. Thanks for starting this interesting discussion.
Second, let me give it a try;
As for the first question, we can check the choices by finding one number which is in the region mentioned but doesn't satisfy our condition:
1) x> 1/2 ; we consider 1; (2|1|-1)/|1|+1 = 1/2 ; thus it won't satisfy the condition, since 1/2 doesn't belong to (1,2).
So the first choice isn't the answer.
2) Again we choose 1 and we reject choice 2
3) Since |-x|=|x| ,and since 1 didn't satisfy the condition; so does -1. If we choose -1 which is less than -1/3, then choice 3 is also false.
So by giving simple examples, we could reject the false choices. It's a good method to check the answers faster. But if we wanna prove that the choice 4 is right then we have:
To simplify the term inside the < < we can write this way: 2|x|-1 = 2(|x|+1) -3 . So we can write: (2|x|-1)/|x|+1 = 2 -3/(|x|+1). Now both sides should be subtracted by 2: -1< (-3)/|x|+1 < 0
Since |x|+1 is never equal to zero, and since it is a positive number, we can multiply all terns by |x|+1 without changing < into > or > into < [Oh, I don't know some of the mathematical terms needed to describe my mean more easily ...Uhhhh]
OK, so we have :
-|x|-1<-3<0 . Since -3<0 is obvious we have : -|x|-1<-3 ; multiplication by -1 gives: |x|+1>3 .So |x|>2 which gives: x>2 or x<-2 CHOICE 4 IS CORRECT
PS- I think you meant : 1< (2|x|-1)/(|x|+1 )< 2; I also checked it without parentheses, but no choice was correct.
W0W!!
Zahra!
How you are good at math!
You surprised me!!
I really impressed by your solution!
Well done, dear…
Well done!
Thank you dear Sahar. That's so kind of you.
I really enjoyed THINKING on a math problem after some months of being away from studies; uh, I need to study more, since university classes are about to start !
Have a nice time
:)
I am majoring in Electrical Engineering, branch of telecommunications .
How about you?
Nice!
What do you think?!
:)
A branch of Math!
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