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
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 ≤ 3We can combine these to write: 3 ≤ x ≤ 3
This means
x MUST equal 3Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: 4^(x + 1) + 4^x > 100Factor 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 ≤ 3We can combine these to write: 3 ≤ x ≤ 3
This means
x MUST equal 3Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: D
Cheers,
Brent
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