Solve the inequality : (0.25)^(x-4)=<(1/16)^x.
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We have to solve: (0.25)^(x-4)=<(1/16)^x
(0.25)^(x-4)=<(1/16)^x
=> (1/4)^(x- 4) =< (1/16)^x
=> (1/4)^(x- 4) =< (1/4)^2x
As (1/4) is less than 1, we have 1/ 4 > (1/4^2)
Similarly, we can say (x- 4) => 2x
=> x - 2x => 4
=> -x => 4
=> x =< -4
Therefore x <= -4.
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We'll transform the decimal form of 0.25 into a fraction:
0.25 = 25/100 = 1/4
We notice that 1/16 = 1/4^2
We'll re-write the inequality:
(1/4)^(x-4) =< (1/4)^2x
Since the bases are below unit, the exponential function is decreasing and we'll get:
x-4 >= 2x
We'll subtract 2x:
x - 2x - 4 >= 0
-x - 4 > = 0
-x >= 4
x =< -4
The range of values of x, that make the inequality to hold, is:
(-infinite ; -4]
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