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5<= 20cos(pi/(20)(x-20))+23<= 20

Solución

5 \leq 20 cos (\frac{π}{20} (x - 20)) + 23 \leq 20

Solución

\frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n \leq x \leq \frac{20 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n or \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n \leq x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

Notación de intervalos

[\frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n, \frac{20 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n] \cup [\frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n, \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n]

Decimal

2.87132 \dots + 40 n \leq x \leq 9.04145 \dots + 40 n or 30.95854 \dots + 40 n \leq x \leq 37.12867 \dots + 40 n

Pasos de solución

5 \leq 20 cos (\frac{π}{20} (x - 20)) + 23 \leq 20

a \leq u \leq b

entonces

a \leq u and u \leq b

5 \leq 20 cos (\frac{π}{20} (x - 20)) + 23 and 20 cos (\frac{π}{20} (x - 20)) + 23 \leq 20

5 \leq 20 cos (\frac{π}{20} (x - 20)) + 23 : \frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n \leq x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

5 \leq 20 cos (\frac{π}{20} (x - 20)) + 23

Intercambiar lados

20 cos (\frac{π}{20} (x - 20)) + 23 \geq 5

Mover

23

al lado derecho

20 cos (\frac{π}{20} (x - 20)) + 23 \geq 5

Restar

23

de ambos lados

20 cos (\frac{π}{20} (x - 20)) + 23 - 23 \geq 5 - 23

Simplificar

20 cos (\frac{π}{20} (x - 20)) \geq - 18

20 cos (\frac{π}{20} (x - 20)) \geq - 18

Dividir ambos lados entre

20

20 cos (\frac{π}{20} (x - 20)) \geq - 18

Dividir ambos lados entre

20

\frac{20 cos ( \frac{π}{20} ( x - 20 ) )}{20} \geq \frac{- 18}{20}

Simplificar

\frac{20 cos ( \frac{π}{20} ( x - 20 ) )}{20} \geq \frac{- 18}{20}

Simplificar

\frac{20 cos ( \frac{π}{20} ( x - 20 ) )}{20} : cos (\frac{π}{20} (x - 20))

\frac{20 cos ( \frac{π}{20} ( x - 20 ) )}{20}

Dividir:

\frac{20}{20} = 1

= cos (\frac{π}{20} (x - 20))

Simplificar

\frac{- 18}{20} : - \frac{9}{10}

\frac{- 18}{20}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{18}{20}

Eliminar los terminos comunes:

2

= - \frac{9}{10}

cos (\frac{π}{20} (x - 20)) \geq - \frac{9}{10}

cos (\frac{π}{20} (x - 20)) \geq - \frac{9}{10}

cos (\frac{π}{20} (x - 20)) \geq - \frac{9}{10}

Para

cos (x) \geq a

, si

- 1 < a < 1

entonces

- arccos (a) + 2 πn \leq x \leq arccos (a) + 2 πn

- arccos (- \frac{9}{10}) + 2 πn \leq \frac{π}{20} (x - 20) \leq arccos (- \frac{9}{10}) + 2 πn

a \leq u \leq b

entonces

a \leq u and u \leq b

- arccos (- \frac{9}{10}) + 2 πn \leq \frac{π}{20} (x - 20) and \frac{π}{20} (x - 20) \leq arccos (- \frac{9}{10}) + 2 πn

- arccos (- \frac{9}{10}) + 2 πn \leq \frac{π}{20} (x - 20) : x \geq \frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

- arccos (- \frac{9}{10}) + 2 πn \leq \frac{π}{20} (x - 20)

Intercambiar lados

\frac{π}{20} (x - 20) \geq - arccos (- \frac{9}{10}) + 2 πn

Multiplicar ambos lados por

20

\frac{π}{20} (x - 20) \geq - arccos (- \frac{9}{10}) + 2 πn

Multiplicar ambos lados por

20

20 \cdot \frac{π}{20} (x - 20) \geq - 20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) \geq - 20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) : π (x - 20)

20 \cdot \frac{π}{20} (x - 20)

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{20 π}{20} (x - 20)

Eliminar los terminos comunes:

20

= (x - 20) π

Simplificar

- 20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn : - 20 arccos (- \frac{9}{10}) + 40 πn

- 20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn

Multiplicar los numeros:

20 \cdot 2 = 40

= - 20 arccos (- \frac{9}{10}) + 40 πn

π (x - 20) \geq - 20 arccos (- \frac{9}{10}) + 40 πn

π (x - 20) \geq - 20 arccos (- \frac{9}{10}) + 40 πn

π (x - 20) \geq - 20 arccos (- \frac{9}{10}) + 40 πn

Dividir ambos lados entre

π

π (x - 20) \geq - 20 arccos (- \frac{9}{10}) + 40 πn

Dividir ambos lados entre

π

\frac{π ( x - 20 )}{π} \geq - \frac{20 arccos ( - \frac{9}{10} )}{π} + \frac{40 πn}{π}

Simplificar

x - 20 \geq - \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n

x - 20 \geq - \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n

Mover

20

al lado derecho

x - 20 \geq - \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n

Sumar

20

a ambos lados

x - 20 + 20 \geq - \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n + 20

Simplificar

x \geq - \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n + 20

x \geq - \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n + 20

Simplificar

- \frac{20 arccos ( - \frac{9}{10} )}{π} + 20 : \frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π}

- \frac{20 arccos ( - \frac{9}{10} )}{π} + 20

Convertir a fracción:

20 = \frac{20 π}{π}

= - \frac{20 arccos ( - \frac{9}{10} )}{π} + \frac{20 π}{π}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π}

x \geq \frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

\frac{π}{20} (x - 20) \leq arccos (- \frac{9}{10}) + 2 πn : x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

\frac{π}{20} (x - 20) \leq arccos (- \frac{9}{10}) + 2 πn

Multiplicar ambos lados por

20

\frac{π}{20} (x - 20) \leq arccos (- \frac{9}{10}) + 2 πn

Multiplicar ambos lados por

20

20 \cdot \frac{π}{20} (x - 20) \leq 20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) \leq 20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) : π (x - 20)

20 \cdot \frac{π}{20} (x - 20)

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{20 π}{20} (x - 20)

Eliminar los terminos comunes:

20

= (x - 20) π

Simplificar

20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn : 20 arccos (- \frac{9}{10}) + 40 πn

20 arccos (- \frac{9}{10}) + 20 \cdot 2 πn

Multiplicar los numeros:

20 \cdot 2 = 40

= 20 arccos (- \frac{9}{10}) + 40 πn

π (x - 20) \leq 20 arccos (- \frac{9}{10}) + 40 πn

π (x - 20) \leq 20 arccos (- \frac{9}{10}) + 40 πn

π (x - 20) \leq 20 arccos (- \frac{9}{10}) + 40 πn

Dividir ambos lados entre

π

π (x - 20) \leq 20 arccos (- \frac{9}{10}) + 40 πn

Dividir ambos lados entre

π

\frac{π ( x - 20 )}{π} \leq \frac{20 arccos ( - \frac{9}{10} )}{π} + \frac{40 πn}{π}

Simplificar

x - 20 \leq \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n

x - 20 \leq \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n

Mover

20

al lado derecho

x - 20 \leq \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n

Sumar

20

a ambos lados

x - 20 + 20 \leq \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n + 20

Simplificar

x \leq \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n + 20

x \leq \frac{20 arccos ( - \frac{9}{10} )}{π} + 40 n + 20

Simplificar

\frac{20 arccos ( - \frac{9}{10} )}{π} + 20 : \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π}

\frac{20 arccos ( - \frac{9}{10} )}{π} + 20

Convertir a fracción:

20 = \frac{20 π}{π}

= \frac{20 arccos ( - \frac{9}{10} )}{π} + \frac{20 π}{π}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π}

x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

Combinar los rangos

x \geq \frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n and x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

Mezclar intervalos sobrepuestos

\frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n \leq x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

20 cos (\frac{π}{20} (x - 20)) + 23 \leq 20 : \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n \leq x \leq \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n

20 cos (\frac{π}{20} (x - 20)) + 23 \leq 20

Mover

23

al lado derecho

20 cos (\frac{π}{20} (x - 20)) + 23 \leq 20

Restar

23

de ambos lados

20 cos (\frac{π}{20} (x - 20)) + 23 - 23 \leq 20 - 23

Simplificar

20 cos (\frac{π}{20} (x - 20)) \leq - 3

20 cos (\frac{π}{20} (x - 20)) \leq - 3

Dividir ambos lados entre

20

20 cos (\frac{π}{20} (x - 20)) \leq - 3

Dividir ambos lados entre

20

\frac{20 cos ( \frac{π}{20} ( x - 20 ) )}{20} \leq \frac{- 3}{20}

Simplificar

cos (\frac{π}{20} (x - 20)) \leq - \frac{3}{20}

cos (\frac{π}{20} (x - 20)) \leq - \frac{3}{20}

Para

cos (x) \leq a

, si

- 1 < a < 1

entonces

arccos (a) + 2 πn \leq x \leq 2 π - arccos (a) + 2 πn

arccos (- \frac{3}{20}) + 2 πn \leq \frac{π}{20} (x - 20) \leq 2 π - arccos (- \frac{3}{20}) + 2 πn

a \leq u \leq b

entonces

a \leq u and u \leq b

arccos (- \frac{3}{20}) + 2 πn \leq \frac{π}{20} (x - 20) and \frac{π}{20} (x - 20) \leq 2 π - arccos (- \frac{3}{20}) + 2 πn

arccos (- \frac{3}{20}) + 2 πn \leq \frac{π}{20} (x - 20) : x \geq \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n

arccos (- \frac{3}{20}) + 2 πn \leq \frac{π}{20} (x - 20)

Intercambiar lados

\frac{π}{20} (x - 20) \geq arccos (- \frac{3}{20}) + 2 πn

Multiplicar ambos lados por

20

\frac{π}{20} (x - 20) \geq arccos (- \frac{3}{20}) + 2 πn

Multiplicar ambos lados por

20

20 \cdot \frac{π}{20} (x - 20) \geq 20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) \geq 20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) : π (x - 20)

20 \cdot \frac{π}{20} (x - 20)

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{20 π}{20} (x - 20)

Eliminar los terminos comunes:

20

= (x - 20) π

Simplificar

20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn : 20 arccos (- \frac{3}{20}) + 40 πn

20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn

Multiplicar los numeros:

20 \cdot 2 = 40

= 20 arccos (- \frac{3}{20}) + 40 πn

π (x - 20) \geq 20 arccos (- \frac{3}{20}) + 40 πn

π (x - 20) \geq 20 arccos (- \frac{3}{20}) + 40 πn

π (x - 20) \geq 20 arccos (- \frac{3}{20}) + 40 πn

Dividir ambos lados entre

π

π (x - 20) \geq 20 arccos (- \frac{3}{20}) + 40 πn

Dividir ambos lados entre

π

\frac{π ( x - 20 )}{π} \geq \frac{20 arccos ( - \frac{3}{20} )}{π} + \frac{40 πn}{π}

Simplificar

x - 20 \geq \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

x - 20 \geq \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

Mover

20

al lado derecho

x - 20 \geq \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

Sumar

20

a ambos lados

x - 20 + 20 \geq \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n + 20

Simplificar

x \geq \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n + 20

x \geq \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n + 20

Simplificar

\frac{20 arccos ( - \frac{3}{20} )}{π} + 20 : \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π}

\frac{20 arccos ( - \frac{3}{20} )}{π} + 20

Convertir a fracción:

20 = \frac{20 π}{π}

= \frac{20 arccos ( - \frac{3}{20} )}{π} + \frac{20 π}{π}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π}

x \geq \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n

\frac{π}{20} (x - 20) \leq 2 π - arccos (- \frac{3}{20}) + 2 πn : x \leq \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n

\frac{π}{20} (x - 20) \leq 2 π - arccos (- \frac{3}{20}) + 2 πn

Multiplicar ambos lados por

20

\frac{π}{20} (x - 20) \leq 2 π - arccos (- \frac{3}{20}) + 2 πn

Multiplicar ambos lados por

20

20 \cdot \frac{π}{20} (x - 20) \leq 20 \cdot 2 π - 20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) \leq 20 \cdot 2 π - 20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn

Simplificar

20 \cdot \frac{π}{20} (x - 20) : π (x - 20)

20 \cdot \frac{π}{20} (x - 20)

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{20 π}{20} (x - 20)

Eliminar los terminos comunes:

20

= (x - 20) π

Simplificar

20 \cdot 2 π - 20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn : 40 π - 20 arccos (- \frac{3}{20}) + 40 πn

20 \cdot 2 π - 20 arccos (- \frac{3}{20}) + 20 \cdot 2 πn

Multiplicar los numeros:

20 \cdot 2 = 40

= 40 π - 20 arccos (- \frac{3}{20}) + 40 πn

π (x - 20) \leq 40 π - 20 arccos (- \frac{3}{20}) + 40 πn

π (x - 20) \leq 40 π - 20 arccos (- \frac{3}{20}) + 40 πn

π (x - 20) \leq 40 π - 20 arccos (- \frac{3}{20}) + 40 πn

Dividir ambos lados entre

π

π (x - 20) \leq 40 π - 20 arccos (- \frac{3}{20}) + 40 πn

Dividir ambos lados entre

π

\frac{π ( x - 20 )}{π} \leq \frac{40 π}{π} - \frac{20 arccos ( - \frac{3}{20} )}{π} + \frac{40 πn}{π}

Simplificar

\frac{π ( x - 20 )}{π} \leq \frac{40 π}{π} - \frac{20 arccos ( - \frac{3}{20} )}{π} + \frac{40 πn}{π}

Simplificar

\frac{π ( x - 20 )}{π} : x - 20

\frac{π ( x - 20 )}{π}

Eliminar los terminos comunes:

π

= x - 20

Simplificar

\frac{40 π}{π} - \frac{20 arccos ( - \frac{3}{20} )}{π} + \frac{40 πn}{π} : 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

\frac{40 π}{π} - \frac{20 arccos ( - \frac{3}{20} )}{π} + \frac{40 πn}{π}

Cancelar

\frac{40 π}{π} : 40

\frac{40 π}{π}

Eliminar los terminos comunes:

π

= 40

= 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + \frac{40 πn}{π}

Cancelar

\frac{40 πn}{π} : 40 n

\frac{40 πn}{π}

Eliminar los terminos comunes:

π

= 40 n

= 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

x - 20 \leq 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

x - 20 \leq 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

x - 20 \leq 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

Mover

20

al lado derecho

x - 20 \leq 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n

Sumar

20

a ambos lados

x - 20 + 20 \leq 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n + 20

Simplificar

x - 20 + 20 \leq 40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n + 20

Simplificar

x - 20 + 20 : x

x - 20 + 20

Sumar elementos similares:

- 20 + 20 \leq 0

= x

Simplificar

40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n + 20 : 40 n + 60 - \frac{20 arccos ( - \frac{3}{20} )}{π}

40 - \frac{20 arccos ( - \frac{3}{20} )}{π} + 40 n + 20

Sumar:

40 + 20 = 60

= 40 n + 60 - \frac{20 arccos ( - \frac{3}{20} )}{π}

x \leq 40 n + 60 - \frac{20 arccos ( - \frac{3}{20} )}{π}

x \leq 40 n + 60 - \frac{20 arccos ( - \frac{3}{20} )}{π}

x \leq 40 n + 60 - \frac{20 arccos ( - \frac{3}{20} )}{π}

Simplificar

60 - \frac{20 arccos ( - \frac{3}{20} )}{π} : \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π}

60 - \frac{20 arccos ( - \frac{3}{20} )}{π}

Convertir a fracción:

60 = \frac{60 π}{π}

= \frac{60 π}{π} - \frac{20 arccos ( - \frac{3}{20} )}{π}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π}

x \leq \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n

Combinar los rangos

x \geq \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n and x \leq \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n

Mezclar intervalos sobrepuestos

\frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n \leq x \leq \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n

Combinar los rangos

\frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n \leq x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n and \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n \leq x \leq \frac{60 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n

Mezclar intervalos sobrepuestos

\frac{- 20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n \leq x \leq \frac{20 π - 20 arccos ( - \frac{3}{20} )}{π} + 40 n or \frac{20 arccos ( - \frac{3}{20} ) + 20 π}{π} + 40 n \leq x \leq \frac{20 arccos ( - \frac{9}{10} ) + 20 π}{π} + 40 n

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