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

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

by Brent@GMATPrepNow » Sun Jun 14, 2020 7:22 am

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BTGModeratorVI wrote: ↑
Sat Jun 13, 2020 5:37 am
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

Answer: D
Source: Manhattan prep

Target question: What is the value of x?

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

Answer: D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pm; Followed by:5 members

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

by deloitte247 » Sun Jun 14, 2020 12:06 pm

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$$4^3=64\ and\ 4^4=256$$
$$for\ 4^x<100\ to\ be\ true\ -3\le x\le3$$
$$possible\ value\ of\ x\ includes\ -3,-2,-1,0,1,2,3$$
$$Statement\ 1\ =>\ 4^{\left(x+1\right)}-4^{\left(x-1\right)}>100$$
$$by\ testing\ all\ possible\ values\ of\ x;$$
$$If\ x\ =-3;\ then\ 4^{\left(x+1\right)}-4^{\left(x-1\right)}>100$$
$$4^{-2}-4^{-4}>100\ \left(this\ is\ false\right)$$
$$16-256>100$$
$$-240<100$$
$$\left(x\ne-3\right)$$
$$If\ x\ =-2;\ then\ 4^{-1}-4^{-3}>100$$
$$-4+64>100\ \left(this\ is\ false\right)$$
$$60<100$$
$$\left(x\ne-2\right)$$
$$If\ x\ =-1;\ then\ 4^0-4^{-2}>100$$
$$1-16>100\ \left(this\ is\ false\right)$$
$$-15<100$$
$$\left(x\ne-1\right)$$
$$If\ x\ =0;\ then\ 4^1-4^{-1}>100$$
$$4+4>100\ \left(this\ is\ false\right)$$
$$8<100$$
$$\left(x\ne0\right)$$
$$If\ x\ =1;\ then\ 4^2-4^0>100$$
$$16-1>100\ \left(this\ is\ false\right)$$
$$15<100$$
$$\left(x\ne1\right)$$ $$If\ x\ =2;\ then\ 4^3-4^1>100$$
$$64-4>100\ \left(this\ is\ false\right)$$
$$60<100$$ $$\left(x\ne2\right)$$
$$If\ x\ =3;\ then\ 4^4-4^2>100$$
$$256-16>100\ \left(this\ is\ true\right)$$
$$240>100$$
$$\left(x=3\right)$$
$$Statement\ 1\ is\ SUFFICIENT$$

$$Statement\ 2=>4^{\left(x+1\right)}+4^x>100$$
If we go through all possible values of x, the value that is true for the information provided in this statement is when x = 3
$$when\ x=3;\ then\ 4^{\left(x+1\right)}+4^x>100$$
$$4^{\left(3+1\right)}+4^3>100$$
$$4^{\left(4\right)}+4^3>100$$
$$256+64>100$$
$$320>100$$
$$this\ is\ true$$
$$statement\ 2\ is\ SUFFICIENT$$
Since each statement alone is SUFFICIENT,
Answer = D

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