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Re: If x is an integer and 4^x < 100, what is x?
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16 Dec 2010, 06:27

4

5

rite2deepti wrote:

If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100 (2) 4^(x + 1) + 4^x > 100 I dont have the OA for this question..I need help ...Thanks

Given: x is an integer and \(4^x < 100\) So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions. \(4^{x - 1}*(4^2 - 1) > 100\) \(4^{x - 1}*15 > 100\) For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\). \(4^x*(4 + 1) > 100\) \(4^x*(5) > 100\) For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).
_________________

Karishma Veritas Prep GMAT Instructor

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Re: If x is an integer and 4^x < 100, what is x?
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16 Dec 2010, 05:08

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rite2deepti wrote:

If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100 (2) 4^(x + 1) + 4^x > 100 I dont have the OA for this question..I need help ...Thanks

answer is D as both statements are sufficient to answer the question.

For 4^x < 100 to hold true X has to be less than Four, X< 4.

(1) 4^(x + 1) – 4^(x – 1) > 100 or it can be simplified to 15 * 4^(x-1) > 100

choose a numbers and work out which number for X will satisfy this statement. (i) choose number 1 answer will be <100 (ii) choose number 2 and answer will be <100 (iii) choose number 3 and answer will be >100 Therefore the the value of X>=3 But for 4^x < 100 to hold true X has to be 3. SUFFICIENT

(2) 4^(x + 1) + 4^x > 100 which can be simplified to 5* 4^x > 100 or to 4^x > 20 choose a numbers and work out which number for X will satisfy this statement. i) choose number 1 answer will be <100 (ii) choose number 2 and answer will be < 100 (iii) choose number 3 and answer will be >100 Therefore the the value of X>=3 But for 4^x < 100 to hold true X has to be 3. SUFFICIENT
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Re: If x is an integer and 4^x < 100, what is x?
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16 Dec 2010, 05:16

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1

If x is an integer and 4^x < 1000, what is x?

Given: \(x=integer\) and \(4^x<1,000\) --> \(2^{2x}<1,000\) --> \(2x<10\) (as \(2^{10}=1024\)) --> \(x<5\), as \(x\) is an integer then it's cam be 4, 3, 2, 1, 0, -1, ...

(1) 4^(x + 1) – 4^(x – 1) > 100 --> factor out \(4^x\) ---> \(4^x(4-\frac{1}{4})>100\) --> \(4^x>\frac{400}{15}\approx{27}\) --> \(x>2\), so \(x\) is either 3 or 4. Not sufficient. (2) 4^(x + 1) + 4^x > 100 --> --> factor out \(4^x\) ---> \(4^x(4+1)>100\) --> \(4^x>20\) --> \(x>2\), so \(x\) is either 3 or 4. Not sufficient.

(1)+(2) The same 2 values for \(x\): 3 and 4. Not sufficient.

Re: If x is an integer and 4^x < 100, what is x?
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03 Nov 2015, 20:55

VeritasPrepKarishma wrote:

rite2deepti wrote:

If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100 (2) 4^(x + 1) + 4^x > 100 I dont have the OA for this question..I need help ...Thanks

Given: x is an integer and \(4^x < 100\) So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions. \(4^{x - 1}*(4^2 - 1) > 100\) \(4^{x - 1}*15 > 100\) For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\). \(4^x*(4 + 1) > 100\) \(4^x*(5) > 100\) For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).

Can someone explain simplifying the equation by factoring out 4^x and so on. I can't seem to wrap my head around it right now.

Re: If x is an integer and 4^x < 100, what is x?
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05 Nov 2015, 00:10

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garey8831 wrote:

Can someone explain simplifying the equation by factoring out 4^x and so on. I can't seem to wrap my head around it right now.

When you have addition/subtraction between terms with exponents, all you can do is take common.

Can you simplify \(4^5 + 4^6\)? Note that \(4^5 = 4*4*4*4*4\) and \(4^6 = 4*4*4*4*4*4\)

\((4*4*4*4*4) + (4*4*4*4*4*4)\)

You can take five 4s common. You will get

\(4*4*4*4*4 ( 1 + 4)\) \(= 4^5 * 5\)

So here, it is the same concept.

\(4^x + 4^{x+1}\) You take \(4^x\) common. \(4^x ( 1 + 4)\) \(4^x * 5\)

and

\(4^{x+1} - 4^{x-1}\) You take \(4^{x-1}\) common because the exponent (x-1) is smaller. \(4^{x-1} ( 4*4 - 1)\) (because \(4^{x+1}\) has two extra 4s so they are left behind} \(4^{x-1} * 15\)
_________________

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Re: If x is an integer and 4^x < 100, what is x?
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02 Sep 2016, 12:39

VeritasPrepKarishma wrote:

rite2deepti wrote:

If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100 (2) 4^(x + 1) + 4^x > 100 I dont have the OA for this question..I need help ...Thanks

Given: x is an integer and \(4^x < 100\) So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions. \(4^{x - 1}*(4^2 - 1) > 100\) \(4^{x - 1}*15 > 100\) For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\). \(4^x*(4 + 1) > 100\) \(4^x*(5) > 100\) For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).

Since the questions just mentions that x is an integer, then why can't x be smaller than -3 such as -4, -5 and so on? Because \(4^{-x} = \frac{1}{4^x}\). Am i missing something here?
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Re: If x is an integer and 4^x < 100, what is x?
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03 Sep 2016, 02:29

1

sam2016 wrote:

VeritasPrepKarishma wrote:

rite2deepti wrote:

If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100 (2) 4^(x + 1) + 4^x > 100 I dont have the OA for this question..I need help ...Thanks

Given: x is an integer and \(4^x < 100\) So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions. \(4^{x - 1}*(4^2 - 1) > 100\) \(4^{x - 1}*15 > 100\) For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\). \(4^x*(4 + 1) > 100\) \(4^x*(5) > 100\) For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater. Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).

Since the questions just mentions that x is an integer, then why can't x be smaller than -3 such as -4, -5 and so on? Because \(4^{-x} = \frac{1}{4^x}\). Am i missing something here?

When you take x as a negative integer, your both the statements A and B would not be satisfied.

Question clearly states that x<4, and both equations prove x=3.

Hence, Answer is D.

Note : try taking the negative values and substitute them in the equations, you will get the equations as incorrect. Hence, Negative values not allowed.
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Re: If x is an integer and 4^x < 100, what is x?
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24 May 2017, 23:27

My method of solving is similar to Bunuel in which we keep the solution terms of 4^x rather than 4^(x-1) as explained by VeritasPrepKarishma. 4^(x-1) maybe difficult for some people to understand but when you solve in 4^x, the expression is same as the one provided in question and makes reasoning and calculation much easier.

Re: If x is an integer and 4^x < 100, what is x?
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24 Jun 2018, 20:18

1

Hi All,

We're told that X is an integer and 4^X < 100. We're asked for the value of X. This question can be solved by TESTing VALUES. There's also a great pattern behind the question that is asked. Since you're told that X is an INTEGER and 4^X < 100, that really means that X is less than or equal to 3.

1) 4^(X + 1) – 4^(X – 1) > 100

Fact 1 gives you a real ugly inequality, so instead of trying to do math, use the basic math that you already know and TEST VALUES...

IF.... If X = 2 ---> 64 - 4 = 60 which is NOT greater than 100, so x CANNOT = 2 If X = 3 ---> 256 - 16 = 240, which IS greater than 100.

There's no point in testing smaller integers, since the total won't be > 100 and you're NOT ALLOWED to go bigger than 3 (because of the information in the prompt). Therefore X must equal 3. Fact 1 is SUFFICIENT

2) 4^(X + 1) + 4^X > 100

Fact 2 provides us with a similar situation, with just a slight change in the math.

IF.... If X = 2 ---> 64 + 16 = 80 which is NOT greater than 100, so X CANNOT = 2 (or anything smaller than 2) If X = 3 ---> 256 + 64 = 320, which IS greater than 100, so X can be 3. X can't be anything other than 3. Fact 2 is SUFFICIENT.

Re: If x is an integer and 4^x < 100, what is x?
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27 Jun 2018, 00:44

Can anyone please explain why cant we consider x=-4 or x=-5 in the first equation. this also satisfies the equation 4^x <100 and we get x=+3,-3,-4,-5........ So statement 1 is not sufficient. Please help.

Re: If x is an integer and 4^x < 100, what is x?
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27 Jun 2018, 03:02

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ambujsrv wrote:

Can anyone please explain why cant we consider x=-4 or x=-5 in the first equation. this also satisfies the equation 4^x <100 and we get x=+3,-3,-4,-5........ So statement 1 is not sufficient. Please help.

If x is an integer and 4^x < 100, what is x?
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29 Dec 2018, 21:44

Testing values definitely looks like the best way to solve this from the responses above. However, if like me the 'what is x' pushed you into trying to calculate x, was there actually a way of doing so? Conversely, any tips to see through that temptation and go straight to testing values?

Re: If x is an integer and 4^x < 100, what is x?
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01 Jan 2019, 04:10

hgfor wrote:

Testing values definitely looks like the best way to solve this from the responses above. However, if like me the 'what is x' pushed you into trying to calculate x, was there actually a way of doing so? Conversely, any tips to see through that temptation and go straight to testing values?

How do you solve 4^x < 100? If x is an integer, you figure out the values of x for which this will hold. Otherwise, you might have to use log tables etc which are not given to you in GMAT.

So when we say x could be 3/2/1/0/-1/-2/-3... (i.e. x <= 3), we have found the values of x for which this works.

Similarly, for statements 1 and 2.
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Karishma Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

Given: x is an integer and 4^x < 100 Let's take a moment to understand what this tells us about the value of INTEGER x 4^3 = 64, and 4^4 = 256 So, if 4^x < 100, then we know that x ≤ 3

Statement 1: 4^(x + 1) – 4^(x – 1) > 100 Factor the left side to get: 4^(x - 1)[4^2 - 1] > 100 Simplify: 4^(x - 1)[15] > 100 Divide both sides by 15 to get: 4^(x - 1) > 100/15 In other words: 4^(x - 1) > 6.66666... 4^0 is NOT greater than 6.6666.... 4^1 is NOT greater than 6.6666.... 4^2 IS greater than 6.6666.... 4^3 IS greater than 6.6666.... . . . etc.

This tells us (x - 1) ≥ 2 Add 1 to both sides to get: x ≥ 3 We also know that x ≤ 3 We can combine these to write: 3 ≤ x ≤ 3 This means x MUST equal 3 Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 4^(x + 1) + 4^x > 100 Factor the left side to get: (4^x)[4^1 - 1] > 100 Simplify: (4^x)[3] > 100 Divide both sides by 3 to get: 4^x > 100/3 In other words: 4^x > 33.333... 4^1 is NOT greater than 33.333... 4^2 is NOT greater than 33.333... 4^3 IS greater than 33.333... 4^4 IS greater than 33.333... 4^5 IS greater than 33.333... . . . etc.

This tells us that x ≥ 3 We also know that x ≤ 3 We can combine these to write: 3 ≤ x ≤ 3 This means x MUST equal 3 Since we can answer the target question with certainty, statement 2 is SUFFICIENT