`e^(-4x)lt=9`

To solve, take the natural logarithm of both sides of the equation.

`ln e^(-4x)lt=ln9`

At the left side, apply the exponent property of logarithm which is `ln a^m=m ln a` .

`-4x ln elt=ln 9`

Note that `ln e =1` . So,

`-4x(1)lt= ln 9`

`-4xlt= ln 9`

Then, divide both sides by -4 to isolate the x.

`(-4x)/(-4) lt= ln9/(-4)`

Since the sign of x changes, the inequality changes too.

`x gt= -ln9/4` **Hence, the solution to the given equation is `xgt=-ln9/4` .**

`e^(-4x)<=9`

Since function `e^(-4x) ` is an one -to-one function, then we can use logartitms:

`-4x<=ln 9`

`x>= -1/4 ln9` for inequality properties.

`x=-0.54930614433405484569762261846126`

Let you see, the value we are searchinfg for , run from the point of graphic with the straight line `y=9` to the right side on. The relative point ,is about `x=-0.54`