What is x*e^{-x}<= 0.01*pi ?

The solution to x*e^{-x} =-\W_{-1}(-0.01pi)

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xe^{-x}<= 0.01pi

x \cdot e^{- x} \leq 0.01 \cdot π

x \leq - W_{0} (- 0.01 π) or x \geq - W_{- 1} (- 0.01 π)

Interval Notation

(- \infty, - W_{0} (- 0.01 π)] \cup [- W_{- 1} (- 0.01 π), \infty)

Decimal Notation

x \leq 0.03245 \dots or x \geq 5.08716 \dots

Solution steps

x e^{- x} \leq 0.01 π

Rewrite the equation with

- x = u

and

x = - u

(- u) e^{u} \leq 0.01 π

Rewrite

(- u) e^{u} \leq 0.01 π

in Lambert form:

e^{u} u \geq - 0.01 π

Solve

e^{u} u \geq - 0.01 π : u \leq W_{- 1} (- 0.01 π) or u \geq W_{0} (- 0.01 π)

u \leq W_{- 1} (- 0.01 π) or u \geq W_{0} (- 0.01 π)

Substitute back

- x = u,

solve for

x

Solve

- x \leq W_{- 1} (- 0.01 π) or (- x) \geq W_{0} (- 0.01 π) : x \leq - W_{0} (- 0.01 π) or x \geq - W_{- 1} (- 0.01 π)

x \leq - W_{0} (- 0.01 π) or x \geq - W_{- 1} (- 0.01 π)

c = (17 (24) + 10)

ln (x^{2} + 10) > ln (x) + ln (7)

\frac{1}{3} + \frac{3}{8}

\frac{2 ^{3} a ^{- 3}}{8 ^{- 1} b ^{- 5} c ^{0}}

\frac{1}{\frac{8}{3}}

What is x*e^{-x}<= 0.01*pi ?
The solution to x*e^{-x}<= 0.01*pi is x<=-\W_{0}(-0.01pi)\lor x>=-\W_{-1}(-0.01pi)