4^(x-1) -2^(6-2x)<10

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вопрос

To solve the inequality 4^(x-1) - 2^(6-2x) < 10, we can simplify the terms using the properties of exponents.

First, we can rewrite 4 as 2^2 and 10 as 2^3. This gives us:

(2^2)^(x-1) - (2^6)/(2^(2x)) < 2^3

Next, we simplify the terms:

2^(2(x-1)) - 2^(6-2x) < 2^3

2^(2x-2) - 2^6 * 2^(-2x) < 2^3

Now, we can combine the terms on the left side of the inequality:

2^(2x-2) - 64 * 2^(-2x) < 8

At this point, we need to choose a common base for the terms. Let's rewrite 64 as 2^6:

2^(2x-2) - 2^6 * 2^(-2x) < 8

Now, we can combine the terms on the left side of the inequality:

2^(2x-2) - 2^(6-2x) < 8

Since the bases are the same, we can equate the exponents:

2x - 2 = 6 - 2x

Solving for x:

2x + 2x = 6 + 2
4x = 8
x = 2

Therefore, the solution to the inequality 4^(x-1) - 2^(6-2x) < 10 is x < 2.