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Popular >

pi/2-arctan(e^x)>0.0001

Solution

\frac{π}{2} - arctan (e^{x}) > 0.0001

Solution

x < ln (9999.99996 \dots)

+2

Interval Notation

(- \infty, ln (9999.99996 \dots))

Decimal Notation

x < 9.21034 \dots

Solution steps

\frac{π}{2} - arctan (e^{x}) > 0.0001

Move

\frac{π}{2}

to the right side

\frac{π}{2} - arctan (e^{x}) > 0.0001

Subtract

\frac{π}{2}

from both sides

\frac{π}{2} - arctan (e^{x}) - \frac{π}{2} > 0.0001 - \frac{π}{2}

Simplify

- arctan (e^{x}) > 0.0001 - \frac{π}{2}

- arctan (e^{x}) > 0.0001 - \frac{π}{2}

Multiply both sides by

- 1

- arctan (e^{x}) > 0.0001 - \frac{π}{2}

Multiply both sides by -1 (reverse the inequality)

(- arctan (e^{x})) (- 1) < 0.0001 (- 1) - \frac{π}{2} (- 1)

Simplify

arctan (e^{x}) < - 0.0001 + \frac{π}{2}

arctan (e^{x}) < - 0.0001 + \frac{π}{2}

If

arctan (x) < a

then

x < tan (a)

e^{x} < tan (- 0.0001 + \frac{π}{2})

tan (- 0.0001 + \frac{π}{2}) = 9999.99996 \dots

tan (- 0.0001 + \frac{π}{2})

tan (- 0.0001 + \frac{π}{2}) = 9999.99996 \dots

= 9999.99996 \dots

e^{x} < 9999.99996 \dots

e^{x} < 9999.99996 \dots : x < ln (9999.99996 \dots)

e^{x} < 9999.99996 \dots

Apply exponent rules

e^{x} < 9999.99996 \dots

Apply exponent rule:

a = b^{l o g_{b} (a)}

9999.99996 \dots = e^{l n (9999.99996 \dots)}

e^{x} < e^{l n (9999.99996 \dots)}

If

a > 1,

then

a^{f (x)} < a^{g (x)}

is equivalent to

f (x) < g (x)

a = e, f (x) = x, g (x) = ln (9999.99996 \dots)

x < ln (9999.99996 \dots)

x < ln (9999.99996 \dots)

x < ln (9999.99996 \dots)

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