2^(x+2) - 2^(x+3) + 5^(x-2) > 5^(x+1) + 2^(x+4)

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First, we can simplify both sides of the inequality:

2^(x+2) - 2^(x+3) + 5^(x-2) > 5^(x+1) + 2^(x+4)

(2^2 2^x) - (2^3 2^x) + (5^x / 5^2) > (5 5^x) + (2^4 2^x)

42^x - 82^x + 5^(x-2) > 55^x + 162^x

-42^x + 5^(x-2) > 55^x + 16*2^x

Next, we can further simplify the inequality by combining like terms and manipulate the inequality to solve for x:

5^(x-2) - 42^x > 55^x + 16*2^x

5^(x-2) - 42^x > 5^x 5 + 2^4 * 2^x

5^(x-2) - 42^x > 5 5^x + 2^x

Now, we can use the properties of exponents to simplify the inequality further. From here, further manipulation of the inequality using exponential properties and algebraic manipulations will help in identifying the range of values of x that satisfy the inequality.