0,6^x<3^x

To solve this inequality, we can start by taking the natural logarithm (ln) of both sides to simplify the expression:

ln(0.6^x) < ln(3^x)

Using the properties of logarithms, we can rewrite this as:

x * ln(0.6) < x * ln(3)

Now, we can divide both sides by x (note that x cannot equal 0 because 0.6^0 = 1 and 3^0 = 1, so the inequality would not hold):

ln(0.6) < ln(3)

Now, we can calculate the natural logarithms of 0.6 and 3, respectively, to see if the inequality holds:

ln(0.6) ≈ -0.5108 ln(3) ≈ 1.0986

Therefore, the inequality ln(0.6) < ln(3) is true, which means that the inequality 0.6^x < 3^x holds for all x where x ≠ 0.