What is (e^{-x})^2-e^{-x}<2 ?

The solution to (e^{-x})^2-e^{-x} -ln(2)

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(e^{-x})^2-e^{-x}<2

(e^{- x})^{2} - e^{- x} < 2

x > - ln (2)

Interval Notation

(- ln (2), \infty)

Decimal Notation

x > - 0.69314 \dots

Solution steps

(e^{- x})^{2} - e^{- x} < 2

Apply exponent rules

(\frac{1}{e ^{x}})^{2} - \frac{1}{e ^{x}} < 2

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{1}{e ^{x}})^{2} e^{x} - \frac{1}{e ^{x}} e^{x} < 2 e^{x}

Simplify

\frac{1}{e ^{x}} - 1 < 2 e^{x}

Let

u = e^{x}

\frac{1}{u} - 1 < 2 u

\frac{1}{u} - 1 < 2 u : - 1 < u < 0 or u > \frac{1}{2}

- 1 < u < 0 or u > \frac{1}{2}

Substitute back

u = e^{x}

- 1 < e^{x} < 0 or e^{x} > \frac{1}{2}

- 1 < e^{x} < 0 :

No Solution

e^{x} > \frac{1}{2} : x > - ln (2)

Combine the intervals

No Solution or x > - ln (2)

Merge Overlapping Intervals

x > - ln (2)

4 m^{2} - 17 m + 18

4 x^{2} + 19 x + 21 = 0

4 x^{2} + 28 x = - 45

10 = 4 (\frac{3 + b}{2})

3 x^{3} - 4 x^{2} - 7 x - 2