x is an integer and x raised to any odd integer is greater

Question

x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^(x-1)-5^x?

(1) z < 25 and w=7^x
(2) x = 4

Bunuel · Answer

x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^(x-1)-5^x?

"x is an integer and x raised to any odd integer is greater than zero" simply means \(x=integer>0\).
The question asks "is w - z greater than 5 times the quantity 7^(x-1)-5^x?".

So, we need to answer whether \(w-z>5(7^{x-1}-5^x)\), given that x is a positive integer.

(1) \(z<25\) and \(w=7^x\)

Is \(7^x-z>5(7^{x-1}-5^x)\)?
Is \(7^x-z>5*7^{x-1}-5^{x+1}\)?
Is \(7^x-5*7^{x-1}+5^{x+1}>z\)?

Since the lowest value of \(x\) is 1, then the lowest value of LHS is when \(x=1\):

\(LHS_{min}=7^x-5*7^{x-1}+5^{x+1}=7-5+25=27\).

So, the lowest value of LHS is 27, which is more than \(z\) as \(z<25\). Therefore, \(7^x-5*7^{x-1}+5^{x+1}>z\) is true. Sufficient.

(2) \(x = 4\). No info about \(w\) and \(z\). Not sufficient.

Answer: A.

_________________

New to the GMAT Club?

Question Bank

Bunuel's Signature Collection

Reading Comprehension

Ultimate Quant Guide

All You Need for Quant

Quantitative

Verbal

Signature Read More

tech3 · Answer

Hello, let me try and explain.

paraphrasing the questions gives us, is w-z > 5.7^(x-1) - 5^(x+1). Further, x is a positive integer as an odd power of x is greater than 0.

taking statement 1, you have the question as 7^x-z > 5.7^(x-1) - 5^(x+1)

Now, for all cases where x is a positive integer, 7^x would be greater than 5.7^(x-1). Next, 5^(x+1) would always be >= 25 as the least value of x can be 1.

So, 7^x > 5.7^x-1
and z < 5^(x+1). Reversing this inequality and adding both the inequalities, we get:

7^x - z > 5.7^x-1 - 5^(x+1),

Hence answer A.

Carcass · Answer

the question is not simple but if you think a bit abstract and test number you can arrive to the solution. The FIRST important thing, breaking the problem, is to understand tha our X is positive. Infact, if a number raised to power of 3 is >0 that means X itself is positive because the odd powr maintain the original sign of the number, so X must be positive. At this point using the exponent rules we have w - z > 5*7^x-1 - 5^x+1 ----> 7^x - 24 (from stem x<25) > 5*7^x-1 - 5^x+1 testing number 1 ------> 7 -24 > 1 - 25 ---> - 17 > -24 this is TRUE. also if you test some value < 25 positive but also negative the result is the same. 1) sufficient x = 4 -------> w - z > 5*7^3 - 5^5..........at this point you know nothing about w and z. 2) insufficient A is the answer.

Kaps07 · Answer

why can't x be zero apart from even ? any odd number raised to the power zero is 1 which is an interger greater than zero. I think x is given as an interger and not only positive integer. if you work with x = 0 , statement 1 is not sufficient ?

Bunuel · Answer

We are told that  integer x raised to to any odd integer is greater than zero :  x^{odd}>0 . Now, if  x=0  then  x^{odd}=0^{odd}=0  (for odd>0), which violates given condition, so  x  cannot be zero.

Hope it's clear.

Kaps07 · Answer

Thanks, I misunderstood the question. I thought it was odd int raised to the power x. It is clear now.

Rajkiranmareedu · Answer

Please explain why the lowest value of X is 1.

Bunuel · Answer

See the highlighted part:  x is an integer and x raised to any odd integer is greater than zero  means that x is a positive integer, thus its lowest possible value is 1.

Hope it's clear.

KarishmaB · Answer

Responding to a pm:

What is the meaning of: x is an integer and x raised to any odd integer is greater than zero?
It is just a convoluted way of saying 'x is a positive integer'. Since x raised to any odd power is positive, x must be positive. x can take values 1, 2, 3, 4 and so on.

Question: Is  w - z > 5*7^{x-1}-5^{x+1}

Using statement 1: Is  7^x - 25 > 5*7^{x-1} - 5^{x+1}

Even though we are given that z is less than 25, let's assume it to be 25. In case we can prove that left hand side is greater than right hand side for z = 25, we can prove that left hand side will remain greater than right hand side for any other value of z. If z has a smaller value, say 9, left had side will become even larger (since a smaller number will be subtracted) than the right hand side.

Now note that  7^x  will be greater than  5*7^{x - 1}  since  7^x  can be written as  7*7^{x - 1}  (x is a positive integer).
Also note that minimum value of x is 1 so  5^{x + 1}  will always be equal to or larger than 25 (x is a positive integer).

So comparing the left and right hand side expressions,  7^x  is greater than  5*7^{x - 1}  and 25 is less than or equal to  5^{x+1} . So left hand side must be greater than the right hand side in all cases when x is a positive integer.

Answer (A)

BrentGMATPrepNow · Answer

Given : x is an integer and x raised to any odd integer is greater than zero KEY CONCEPT: A number raised to an ODD power retains its sign. That is, POSITIVE^ODD = POSITIVE and NEGATIVE^ODD = NEGATIVE The given info tells us that x^ ODD = POSITIVE So, it must be the case that x is a POSITIVE integer Target question : Is w - z > 5 Let's tidy the target question by expanding the right side of the inequality to get.... REPHRASED target question : Is w - z > 5 - 5^(x+1)? Statement 1 : z < 25 and w = 7^x Take the REPHRASED target question and replace w with 7^x to get: Is 7^x - z > 5 - 5^(x+1)? IMPORTANT: Notice that, on the left side of the inequality, we're subtracting z We're told that z < 25 So, we can help MINIMIZE the value of the left side by making z as big as possible So, let's make z = 25 (aside: we could make z equal something really close to 25, like 24.99999999999, but let's make things easy on ourselves and make z EQUAL 25. We get: Is 7^x - 25 > 5 - 5^(x+1)? Now let's get like terms on the same side of the inequality. First subtract 5 from both sides to get: Is 7^x - 5 - 25 > -5^(x+1)? Then add 25 to both sides to get: Is 7^x - 5 > 25 - 5^(x+1)? Factor 7^(x-1) from left side to get: Is 7^(x-1) > 25 - 5^(x+1)? Simplify left side to get: Is 7^(x-1) > 25 - 5^(x+1)? Now recognize that 7^(x-1) will be POSITIVE for all values of x. So, 7^(x-1) must be POSITIVE Now recognize that since x is a POSITIVE integer , 5^(x+1) must be greater than or equal to 25 (since x = 1, is the smallest possible value of x) So, 25 - 5^(x+1) must be less than or equal to zero. So, our question becomes Is some POSITIVE number > some number that's less than or equal to zero? The answer to this REPHRASED target question is a definitive YES Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT Statement 2 : x = 4 Since we have no information about z or w, we cannot answer the REPHRASED target question with certainty. So, statement 2 is NOT SUFFICIENT Answer: A Cheers, Brent

Kimberly77 · Answer

Thanks Brent  BrentGMATPrepNow  for detaailed explanation, To clarify this part of 5^x > 5^(x+1), Is it 5 times the quantity 5(7^(x-1)-5^x)? Therefore we've 1 extra 5 here when equation expanded ? Thanks Brent

Target question: Is w - z > 5 
Let's tidy the target question by expanding the right side of the inequality to get....
REPHRASED target question: Is w - z > 5  - 5^(x+1)?

BrentGMATPrepNow · Answer

Thanks Brent  BrentGMATPrepNow  for detaailed explanation, To clarify this part of 5^x > 5^(x+1), Is it 5 times the quantity 5(7^(x-1)-5^x)? Therefore we've 1 extra 5 here when equation expanded ? Thanks Brent

Target question: Is w - z > 5 
Let's tidy the target question by expanding the right side of the inequality to get....
REPHRASED target question: Is w - z > 5  - 5^(x+1)?

Key property: a(b + c) = ab + ac
In the expression  , we have two terms: 7^(x-1) and 5^x. 
You can consider these terms b and c (as in the property a(b + c) = ab + ac)

In the expression 5 , I am multiplying each of the two terms by 5. 
When we do so we get: 5  - 5(5^x)
If we focus on the second part, 5(5^x), we get: 
5(5^x) = (5^1)(5^x) = 5^(x+1)

Does that help?

Kimberly77 · Answer

Great explanation always !! Thanks  BrentGMATPrepNow

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

x is an integer and x raised to any odd integer is greater

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia