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Solving inequalities/ sin(4*x)/3>-3/2

sin(4*x)/3>-3/2 inequation

A inequation with variable

The solution

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sin(4*x)       
-------- > -3/2
   3
sin⁡(4x)3>−32\frac{\sin{\left(4 x \right)}}{3} > - \frac{3}{2} 
sin(4*x)/3 > -3/2
Detail solution
Given the inequality:
sin⁡(4x)3>−32\frac{\sin{\left(4 x \right)}}{3} > - \frac{3}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(4x)3=−32\frac{\sin{\left(4 x \right)}}{3} = - \frac{3}{2}
Solve:
Given the equation
sin⁡(4x)3=−32\frac{\sin{\left(4 x \right)}}{3} = - \frac{3}{2}
- this is the simplest trigonometric equation
Divide both parts of the equation by 1/3

The equation is transformed to
sin⁡(4x)=−92\sin{\left(4 x \right)} = - \frac{9}{2}
As right part of the equation
modulo =
True

but sin
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
x1=π4+asin⁡(92)4x_{1} = \frac{\pi}{4} + \frac{\operatorname{asin}{\left(\frac{9}{2} \right)}}{4}
x2=−asin⁡(92)4x_{2} = - \frac{\operatorname{asin}{\left(\frac{9}{2} \right)}}{4}
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0

sin⁡(0⋅4)3>−32\frac{\sin{\left(0 \cdot 4 \right)}}{3} > - \frac{3}{2}
0 > -3/2

so the inequality is always executed
Solving inequality on a graph
02468-8-6-4-2-10102-2
Rapid solution
This inequality holds true always
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