(9/25)*(5/3)^(x-2)>= корень(27/125)^х

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вопрос

To solve this inequality, we need to simplify both sides of the inequality:

Left side: (9/25) (5/3)^(x-2)
= (9/25) (5^(x-2) / 3^(x-2))
= (9/25) 5^(x-2) / 3^(x-2)
= 9/25 5^x / 5^2 / 3^(x-2)
= 9/25 5^x / 25 / 3^(x-2)
= 9/25 5^x / 25 / 3^x 3^2
= 9/25 (1/5)^x / 3^2
= 9 / (25 5^x) / 9
= 1 / (25 5^x)

Right side: sqrt((27/125)^x)
= sqrt(3^3 / 5^3)^x
= sqrt(3^(3x) / 5^(3x))
= sqrt(3^(3x)) / sqrt(5^(3x))
= 3^(3x/2) / 5^(3x/2)
= (3/5)^(3x/2)

Therefore, the inequality becomes:

1 / (25 * 5^x) >= (3/5)^(3x/2)

To further simplify, we can rewrite the right side as:

1 / (25 5^x) >= (3/5)^(3x/2)
25 5^x <= 5^(3x/2)
5^(1 + x) <= 5^(3x/2)
5^(1/2) <= 5^(3x/2 - x)
5^(1/2 + x) <= 5^(3/2)
1/2 + x <= 3/2
x <= 1

Therefore, the solution to the inequality is x <= 1.