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 <= 3/2x <= 1
Therefore, the solution to the inequality is x <= 1.
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.