Hi Cazubuine,
This DS question is great because it has a "question behind the question." 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. The prompt asks for the value of x? You do NOT want to try to solve this algebraically.
1) 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 IT.
Plugging in values for x, you'll find...
If x = 2 ---> total = 60 which is NOT greater than 100, so x CANNOT = 2
If x = 3 ---> total = 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 = 3. Fact 1 is SUFFICIENT
2) Fact 2 provides you a similar situation, with just a slight change in the math.
Plug in values here and you'll find that X MUST = 3.
Fact 2 is SUFFICIENT.
Final Answer: D (Both are SUFFICIENT).
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Plugging in values seems easier, but here's an algebraic approach.cazubuine wrote:Can this be solved algebraically:
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 [/b]
Statement 1: 4^(x + 1) - 4^(x - 1) > 100
Factor out 4^x:
4^x * (4¹ - 4¯¹) > 100
Since (4¹ - 4¯¹) = 4 - 1/4 ≈ 4, we can ballpark:
4^x * 4 > 100
4^x > 25.
Thus, 25 < 4^x < 100, implying that x=3.
SUFFICIENT.
Statement 2: 4^(x + 1) + 4^x > 100
Factor out 4^x:
4^x * (4¹ + 1) > 100
4^x * 5 > 100
4^x > 20.
Thus, 20 < 4^x < 100, implying that x=3.
SUFFICIENT.
The correct answer is D.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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