Exponents and In-equalities

gmattesttaker2 wrote:Hello,

This problem is taken from MGMAT Strategy Guide 2 (5th edition), P. 125. Can you please assist here?

Determine whether True or False:

2) ( (x+1)/x )^-2 > (x+1)/x , if x>0

( (x+1)/x )^-2 = ( x/(x+1) )^2

If x>0 , then x+1>x

Hence (x+1)/x>1 and 0<x/(x+1)<1

Now, we know that square of a number (y), where 0<y<1, is always less than the number(y).
=>( x/(x+1) )^2< ( x/(x+1) )< (x+1)/x.

Therefore the statement is false

gmattesttaker2 wrote:Hello,

This problem is taken from MGMAT Strategy Guide 2 (5th edition), P. 125. Can you please assist here?

Determine whether True or False:

2) ( (x+1)/x )^-2 > (x+1)/x , if x>0

[spoiler]Ans: False[/spoiler]

Looking at the answer given in the book, I tried plugging in 0.2 for x and was able to get to the answer. However, I was just wondering if this can be solved without plugging in any values? Thanks for your help.

Best Regards,
Sri

Hi Sri:

Algebraic solution:
.Let's simplify this expression assuming it is true:

( (x+1)/x )^-2 > (x+1)/x
=> (x/x+1)^2 > (x+1)/x
=> 1> ((x+1)/x)^3 (Multiplying both sides by ((x+1)/x)^2 since it is positive)
=> 1 > (1+1/x)^3

Now we know that x>0 therefore 1/x > 0 => 1+1/x > 1
=> (1+1/x)^3 > 1. Which is opposite of what we simplified the above expression to:

Hence the given expression is FALSE.

Even though you can solve it this way, I would suggest just checking with x = 1 to see whether it is true.

Cheers!

niketdoshi123 wrote:

gmattesttaker2 wrote:Hello,

This problem is taken from MGMAT Strategy Guide 2 (5th edition), P. 125. Can you please assist here?

Determine whether True or False:

2) ( (x+1)/x )^-2 > (x+1)/x , if x>0

( (x+1)/x )^-2 = ( x/(x+1) )^2

If x>0 , then x+1>x

Hence (x+1)/x>1 and 0<x/(x+1)<1

Now, we know that square of a number (y), where 0<y<1, is always less than the number(y).
=>( x/(x+1) )^2< ( x/(x+1) )< (x+1)/x.

Therefore the statement is false

Hello Niket,

Thank you very much for the explanation.

Best Regards,
Sri

eagleeye wrote:

gmattesttaker2 wrote:Hello,

This problem is taken from MGMAT Strategy Guide 2 (5th edition), P. 125. Can you please assist here?

Determine whether True or False:

2) ( (x+1)/x )^-2 > (x+1)/x , if x>0

[spoiler]Ans: False[/spoiler]

Looking at the answer given in the book, I tried plugging in 0.2 for x and was able to get to the answer. However, I was just wondering if this can be solved without plugging in any values? Thanks for your help.

Best Regards,
Sri

Hi Sri:

Algebraic solution:
.Let's simplify this expression assuming it is true:

( (x+1)/x )^-2 > (x+1)/x
=> (x/x+1)^2 > (x+1)/x
=> 1> ((x+1)/x)^3 (Multiplying both sides by ((x+1)/x)^2 since it is positive)
=> 1 > (1+1/x)^3

Now we know that x>0 therefore 1/x > 0 => 1+1/x > 1
=> (1+1/x)^3 > 1. Which is opposite of what we simplified the above expression to:

Hence the given expression is FALSE.

Even though you can solve it this way, I would suggest just checking with x = 1 to see whether it is true.

Cheers!

Hello Eagleeye,

Thank you very much for the thorough explanation.

Best Regards,
Sri