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!

.

.

.

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 rightthen 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 < 0Since |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 CORRECTPS-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!

Next