7*(2^(2x)) + 2^(2x+1) < 3^(2x+1) + 3^(2x)

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To solve this inequality, let's first simplify both sides:

7*(2^(2x)) + 2^(2x+1) < 3^(2x+1) + 3^(2x)

Expanding the terms:

72^(2x) + 22^(2x) < 3*3^(2x) + 3^(2x)

Now we can simplify the exponents using the property a^(m+n) = a^m * a^n:

72^(2x) + 222^2x < 33*3^(2x) + 3^(2x)

Simplify the exponents:

7(2^x)^2 + 22^(x+1) < 3*(3^x)^2 + 3^x

Let's simplify further:

7(2^x)^2 + 22^(x+1) < 3(3^x)^2 + 3^x
74^x + 222^x < 39^x + 3^x
74^x + 4*2^x < 27^x + 3^x

Now, we see that we have a mixture of base 2, base 3, and base 4 numbers. To make further simplification, we can note that 4 = 2^2 and 27 = 3^3:

7(2^x)^2 + 42^x < 3(3^x)^2 + 3^x
7(2^x)^2 + 4(2^x)^2 < 3(3^x)^2 + 3^x
(7+4)(2^x)^2 < (3+1)(3^x)^2
11(2^x)^2 < 4(3^x)^2
112^(2x) < 43^(2x)

This is the simplified version of the inequality. We can see that the inequality is dependent on the values of x, and further steps may require numerical approximations.