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

Solution

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

Solution

\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

Interval Notation

[\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 Notation

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

Solution steps

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

a \leq u \leq b

then

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

Switch sides

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

Move

23

to the right side

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

Subtract

23

from both sides

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

Simplify

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

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

Divide both sides by

20

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

Divide both sides by

20

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

Simplify

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

Simplify

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

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

Divide the numbers:

\frac{20}{20} = 1

= cos (\frac{π}{20} (x - 20))

Simplify

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

\frac{- 18}{20}

Apply the fraction rule:

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

= - \frac{18}{20}

Cancel the common factor:

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}

For

cos (x) \geq a

, if

- 1 < a < 1

then

- 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

then

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)

Switch sides

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

Multiply both sides by

20

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

Multiply both sides by

20

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

Simplify

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

Simplify

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

20 \cdot \frac{π}{20} (x - 20)

Multiply fractions:

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

= \frac{20 π}{20} (x - 20)

Cancel the common factor:

20

= (x - 20) π

Simplify

- 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

Multiply the numbers:

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

Divide both sides by

π

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

Divide both sides by

π

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

Simplify

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

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

Move

20

to the right side

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

Add

20

to both sides

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

Simplify

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

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

Simplify

- \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

Convert element to fraction:

20 = \frac{20 π}{π}

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

Since the denominators are equal, combine the fractions:

\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

Multiply both sides by

20

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

Multiply both sides by

20

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

Simplify

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

Simplify

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

20 \cdot \frac{π}{20} (x - 20)

Multiply fractions:

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

= \frac{20 π}{20} (x - 20)

Cancel the common factor:

20

= (x - 20) π

Simplify

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

Multiply the numbers:

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

Divide both sides by

π

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

Divide both sides by

π

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

Simplify

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

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

Move

20

to the right side

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

Add

20

to both sides

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

Simplify

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

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

Simplify

\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

Convert element to fraction:

20 = \frac{20 π}{π}

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

Since the denominators are equal, combine the fractions:

\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

Combine the intervals

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

Merge Overlapping Intervals

\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

Move

23

to the right side

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

Subtract

23

from both sides

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

Simplify

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

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

Divide both sides by

20

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

Divide both sides by

20

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

Simplify

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

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

For

cos (x) \leq a

, if

- 1 < a < 1

then

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

then

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)

Switch sides

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

Multiply both sides by

20

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

Multiply both sides by

20

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

Simplify

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

Simplify

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

20 \cdot \frac{π}{20} (x - 20)

Multiply fractions:

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

= \frac{20 π}{20} (x - 20)

Cancel the common factor:

20

= (x - 20) π

Simplify

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

Multiply the numbers:

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

Divide both sides by

π

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

Divide both sides by

π

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

Simplify

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

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

Move

20

to the right side

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

Add

20

to both sides

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

Simplify

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

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

Simplify

\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

Convert element to fraction:

20 = \frac{20 π}{π}

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

Since the denominators are equal, combine the fractions:

\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

Multiply both sides by

20

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

Multiply both sides by

20

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

Simplify

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

Simplify

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

20 \cdot \frac{π}{20} (x - 20)

Multiply fractions:

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

= \frac{20 π}{20} (x - 20)

Cancel the common factor:

20

= (x - 20) π

Simplify

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

Multiply the numbers:

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

Divide both sides by

π

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

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π ( x - 20 )}{π} : x - 20

\frac{π ( x - 20 )}{π}

Cancel the common factor:

π

= x - 20

Simplify

\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}

Cancel

\frac{40 π}{π} : 40

\frac{40 π}{π}

Cancel the common factor:

π

= 40

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

Cancel

\frac{40 πn}{π} : 40 n

\frac{40 πn}{π}

Cancel the common factor:

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

Move

20

to the right side

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

Add

20

to both sides

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

Simplify

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

Simplify

x - 20 + 20 : x

x - 20 + 20

Add similar elements:

- 20 + 20 \leq 0

= x

Simplify

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

Add the numbers:

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}

Simplify

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}

Convert element to fraction:

60 = \frac{60 π}{π}

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

Since the denominators are equal, combine the fractions:

\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

Combine the intervals

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

Merge Overlapping Intervals

\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

Combine the intervals

\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

Merge Overlapping Intervals

\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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