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]
Plugging in values seems easier, but here's an algebraic approach.
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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