What is x/(e^x)>=-0.2679 ?

The solution to x/(e^x)>=-0.2679 is x>=-\W_{0}(0.2679)

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x/(e^x)>=-0.2679

\frac{x}{e ^{x}} \geq - 0.2679

x \geq - W_{0} (0.2679)

Interval Notation

[- W_{0} (0.2679), \infty)

Decimal Notation

x \geq - 0.21588 \dots

Solution steps

\frac{x}{e ^{x}} \geq - 0.2679

f (x) > 0

we can multiply or devide both sides of inequality by

f (x)

e^{x}

is greater than

0

for all

x

\frac{x}{e ^{x}} e^{x} \geq - 0.2679 e^{x}

Simplify

x \geq - 0.2679 e^{x}

Prepare

x \geq - 0.2679 e^{x}

for Lambert form:

x e^{- x} \geq - 0.2679

Rewrite the equation with

- x = u

and

x = - u

(- u) e^{u} \geq - 0.2679

Rewrite

(- u) e^{u} \geq - 0.2679

in Lambert form:

e^{u} u \leq 0.2679

Solve

e^{u} u \leq 0.2679 : u \leq W_{0} (0.2679)

u \leq W_{0} (0.2679)

Substitute back

- x = u,

solve for

x

Solve

- x \leq W_{0} (0.2679) : x \geq - W_{0} (0.2679)

x \geq - W_{0} (0.2679)

u^{2} + 2 u + 1 = 0

\frac{5 - x}{2} - 2 = \frac{1 - x}{2} - \frac{2 ( x + 1 )}{3}

(x - 3)^{2} (12 - 3 x) > 0

\frac{8 ^{- 6}}{4 ^{- 6}}

0 < \frac{1}{3} x < π

What is x/(e^x)>=-0.2679 ?
The solution to x/(e^x)>=-0.2679 is x>=-\W_{0}(0.2679)