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3^t-4t-1<0

Solución

3^{t} - 4 t - 1 < 0

Solución

t < - \frac{W _{0} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4} or t > - \frac{W _{- 1} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4}

+2

Notación de intervalos

- \infty, - \frac{W _{0} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4} \cup - \frac{W _{- 1} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4}, \infty

Notación decimal

t < 0 or t > 2

Pasos de solución

3^{t} - 4 t - 1 < 0

Mover

4 t

al lado derecho

3^{t} - 1 < 4 t

Mover

1

al lado derecho

3^{t} < 4 t + 1

Preparar

3^{t} < 4 t + 1

para la forma de Lambert:

1 < (4 t + 1) e^{- l n (3) t}

Re-escribir la ecuación con

(- t - \frac{1}{4}) ln (3) = u

y

t = - \frac{4 u + ln ( 3 )}{4 ln ( 3 )}

1 < (4 (- \frac{4 u + ln ( 3 )}{4 ln ( 3 )}) + 1) e^{- l n (3) (- \frac{4 u + l n ( 3 )}{4 l n ( 3 )})}

Simplificar

1 < e^{\frac{4 u + l n ( 3 )}{4}} (- \frac{4 u + ln ( 3 )}{ln ( 3 )} + 1)

Reescribir

1 < e^{\frac{4 u + l n ( 3 )}{4}} (- \frac{4 u + ln ( 3 )}{ln ( 3 )} + 1)

en forma de Lambert:

\frac{e ^{\frac{4 u + l n ( 3 )}{4}} u}{e ^{\frac{l n ( 3 )}{4}}} > - \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}}

Resolver

\frac{e ^{\frac{4 u + l n ( 3 )}{4}} u}{e ^{\frac{l n ( 3 )}{4}}} > - \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} : u < W_{- 1} (- \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}}) or u > W_{0} (- \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}})

u < W_{- 1} (- \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}}) or u > W_{0} (- \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}})

Sustituir hacia atrás la

(- t - \frac{1}{4}) ln (3) = u,

resolver para

t

Resolver

(- t - \frac{1}{4}) ln (3) < W_{- 1} (- \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}}) or (- t - \frac{1}{4}) ln (3) > W_{0} (- \frac{ln ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}}) : t < - \frac{W _{0} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4} or t > - \frac{W _{- 1} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4}

t < - \frac{W _{0} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4} or t > - \frac{W _{- 1} ( - \frac{l n ( 3 )}{4 e ^{\frac{l n ( 3 )}{4}}} )}{ln ( 3 )} - \frac{1}{4}

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