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Popular >

5/4 sin(3/4 x-6)+1/4 <= 3/2

Solution

\frac{5}{4} sin (\frac{3}{4} x - 6) + \frac{1}{4} \leq \frac{3}{2}

Solution

True for all x \in R

Interval Notation

(- \infty, \infty)

Solution steps

\frac{5}{4} sin (\frac{3}{4} x - 6) + \frac{1}{4} \leq \frac{3}{2}

Move

\frac{1}{4}

to the right side

\frac{5}{4} sin (\frac{3}{4} x - 6) + \frac{1}{4} \leq \frac{3}{2}

Subtract

\frac{1}{4}

from both sides

\frac{5}{4} sin (\frac{3}{4} x - 6) + \frac{1}{4} - \frac{1}{4} \leq \frac{3}{2} - \frac{1}{4}

Simplify

\frac{5}{4} sin (\frac{3}{4} x - 6) + \frac{1}{4} - \frac{1}{4} \leq \frac{3}{2} - \frac{1}{4}

Simplify

\frac{5}{4} sin (\frac{3}{4} x - 6) + \frac{1}{4} - \frac{1}{4} : \frac{5}{4} sin (\frac{3}{4} x - 6)

\frac{5}{4} sin (\frac{3}{4} x - 6) + \frac{1}{4} - \frac{1}{4}

Add similar elements:

\frac{1}{4} - \frac{1}{4} \leq 0

= \frac{5}{4} sin (\frac{3}{4} x - 6)

Simplify

\frac{3}{2} - \frac{1}{4} : \frac{5}{4}

\frac{3}{2} - \frac{1}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{3}{2} :

multiply the denominator and numerator by

2

\frac{3}{2} = \frac{3 \cdot 2}{2 \cdot 2} = \frac{6}{4}

= \frac{6}{4} - \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{6 - 1}{4}

Subtract the numbers:

6 - 1 = 5

= \frac{5}{4}

\frac{5}{4} sin (\frac{3}{4} x - 6) \leq \frac{5}{4}

\frac{5}{4} sin (\frac{3}{4} x - 6) \leq \frac{5}{4}

\frac{5}{4} sin (\frac{3}{4} x - 6) \leq \frac{5}{4}

Multiply both sides by

4

\frac{5}{4} sin (\frac{3}{4} x - 6) \leq \frac{5}{4}

Multiply both sides by

4

4 \cdot \frac{5}{4} sin (\frac{3}{4} x - 6) \leq \frac{5 \cdot 4}{4}

Simplify

4 \cdot \frac{5}{4} sin (\frac{3}{4} x - 6) \leq \frac{5 \cdot 4}{4}

Simplify

4 \cdot \frac{5}{4} sin (\frac{3}{4} x - 6) : 5 sin (\frac{3}{4} x - 6)

4 \cdot \frac{5}{4} sin (\frac{3}{4} x - 6)

Multiply fractions:

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

= \frac{5 \cdot 4}{4} sin (\frac{3}{4} x - 6)

Cancel the common factor:

4

= sin (\frac{3}{4} x - 6) \cdot 5

Simplify

\frac{5 \cdot 4}{4} : 5

\frac{5 \cdot 4}{4}

Multiply the numbers:

5 \cdot 4 = 20

= \frac{20}{4}

Divide the numbers:

\frac{20}{4} = 5

= 5

5 sin (\frac{3}{4} x - 6) \leq 5

5 sin (\frac{3}{4} x - 6) \leq 5

5 sin (\frac{3}{4} x - 6) \leq 5

Divide both sides by

5

5 sin (\frac{3}{4} x - 6) \leq 5

Divide both sides by

5

\frac{5 sin ( \frac{3}{4} x - 6 )}{5} \leq \frac{5}{5}

Simplify

sin (\frac{3}{4} x - 6) \leq 1

sin (\frac{3}{4} x - 6) \leq 1

Range of

sin (\frac{3}{4} x - 6) : - 1 \leq sin (\frac{3}{4} x - 6) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (\frac{3}{4} x - 6) \leq 1

- 1 \leq sin (\frac{3}{4} x - 6) \leq 1

sin (\frac{3}{4} x - 6) \leq 1 and - 1 \leq sin (\frac{3}{4} x - 6) \leq 1 : - 1 \leq sin (\frac{3}{4} x - 6) \leq 1

Let

y = sin (\frac{3}{4} x - 6)

Combine the intervals

y \leq 1 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq 1 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \leq 1

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

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