To solve these inequalities, we will first simplify the expressions on the left side of each inequality:
(5^{x-1} < 25)[5^{x-1} < 25][5^{x-1} < 5^2][x-1 < 2][x < 3]
(3^{3-x} > 9)[3^{3-x} > 9][3^{3-x} > 3^2][3-x > 2][-x > -1][x < 1]
(6^{2x} < \frac{1}{36})[6^{2x} < \frac{1}{36}][(6^{2})^x < 6^{-2}][36^x < \frac{1}{36}]Since 36 is (6^2), we can rewrite the inequality as:[6^2x < 6^{-2}][6^2x < \frac{1}{6^2}][6^2x < \frac{1}{36}][2x < -2][x < -1]
Therefore, the solutions to the inequalities are:
To solve these inequalities, we will first simplify the expressions on the left side of each inequality:
(5^{x-1} < 25)
[5^{x-1} < 25]
[5^{x-1} < 5^2]
[x-1 < 2]
[x < 3]
(3^{3-x} > 9)
[3^{3-x} > 9]
[3^{3-x} > 3^2]
[3-x > 2]
[-x > -1]
[x < 1]
(6^{2x} < \frac{1}{36})
[6^{2x} < \frac{1}{36}]
[(6^{2})^x < 6^{-2}]
[36^x < \frac{1}{36}]
Since 36 is (6^2), we can rewrite the inequality as:
[6^2x < 6^{-2}]
[6^2x < \frac{1}{6^2}]
[6^2x < \frac{1}{36}]
[2x < -2]
[x < -1]
Therefore, the solutions to the inequalities are:
(x < 3)(x < 1)(x < -1)