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Popular >

(2+sqrt(3)sin(x))<0

Solution

(2 + 3 sin (x)) < 0

False for all x \in R

Solution steps

2 + 3 sin (x) < 0

Move

2

to the right side

2 + 3 sin (x) < 0

Subtract

2

from both sides

2 + 3 sin (x) - 2 < 0 - 2

Simplify

3 sin (x) < - 2

3 sin (x) < - 2

Divide both sides by

3

3 sin (x) < - 2

Divide both sides by

3

\frac{3 sin ( x )}{3} < \frac{- 2}{3}

Simplify

\frac{3 sin ( x )}{3} < \frac{- 2}{3}

Simplify

\frac{3 sin ( x )}{3} : sin (x)

\frac{3 sin ( x )}{3}

Cancel the common factor:

3

= sin (x)

Simplify

\frac{- 2}{3} : - \frac{2 3}{3}

\frac{- 2}{3}

Apply the fraction rule:

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

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

sin (x) < - \frac{2 3}{3}

sin (x) < - \frac{2 3}{3}

sin (x) < - \frac{2 3}{3}

Range of

sin (x) : - 1 \leq sin (x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (x) \leq 1

- 1 \leq sin (x) \leq 1

sin (x) < - \frac{2 3}{3} and - 1 \leq sin (x) \leq 1 :

False

Let

y = sin (x)

Combine the intervals

y < - \frac{2 3}{3} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y < - \frac{2 3}{3} and - 1 \leq y \leq 1

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

y < - \frac{2 3}{3}

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

2 cos (x) - 1 > 0

sin (2 x) - cos (2 x) > 0

8 > cos (θ), < - 9, - 4 > \cdot < - 9

cos (2 x) + 0.5 > 0

tan (x) - \frac{3}{3} \leq 0