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Solving inequalities/ sin2x+sinx>0

sin2x+sinx>0 inequation

A inequation with variable

The solution

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sin(2*x) + sin(x) > 0
sin⁡(x)+sin⁡(2x)>0\sin{\left(x \right)} + \sin{\left(2 x \right)} > 0 
sin(x) + sin(2*x) > 0
Detail solution
Given the inequality:
sin⁡(x)+sin⁡(2x)>0\sin{\left(x \right)} + \sin{\left(2 x \right)} > 0
To solve this inequality, we must first solve the corresponding equation:
sin⁡(x)+sin⁡(2x)=0\sin{\left(x \right)} + \sin{\left(2 x \right)} = 0
Solve:
x1=0x_{1} = 0
x2=−4π3x_{2} = - \frac{4 \pi}{3}
x3=−πx_{3} = - \pi
x4=−2π3x_{4} = - \frac{2 \pi}{3}
x5=2π3x_{5} = \frac{2 \pi}{3}
x6=πx_{6} = \pi
x7=4π3x_{7} = \frac{4 \pi}{3}
x8=2πx_{8} = 2 \pi
x1=0x_{1} = 0
x2=−4π3x_{2} = - \frac{4 \pi}{3}
x3=−πx_{3} = - \pi
x4=−2π3x_{4} = - \frac{2 \pi}{3}
x5=2π3x_{5} = \frac{2 \pi}{3}
x6=πx_{6} = \pi
x7=4π3x_{7} = \frac{4 \pi}{3}
x8=2πx_{8} = 2 \pi
This roots
x2=−4π3x_{2} = - \frac{4 \pi}{3}
x3=−πx_{3} = - \pi
x4=−2π3x_{4} = - \frac{2 \pi}{3}
x1=0x_{1} = 0
x5=2π3x_{5} = \frac{2 \pi}{3}
x6=πx_{6} = \pi
x7=4π3x_{7} = \frac{4 \pi}{3}
x8=2πx_{8} = 2 \pi
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x2x_{0} < x_{2}
For example, let's take the point
x0=x2−110x_{0} = x_{2} - \frac{1}{10}
=
−4π3−110- \frac{4 \pi}{3} - \frac{1}{10}
=
−4π3−110- \frac{4 \pi}{3} - \frac{1}{10}
substitute to the expression
sin⁡(x)+sin⁡(2x)>0\sin{\left(x \right)} + \sin{\left(2 x \right)} > 0
sin⁡(2(−4π3−110))+sin⁡(−4π3−110)>0\sin{\left(2 \left(- \frac{4 \pi}{3} - \frac{1}{10}\right) \right)} + \sin{\left(- \frac{4 \pi}{3} - \frac{1}{10} \right)} > 0
     /1   pi\      /1    pi\    
- cos|- + --| + sin|-- + --| > 0
     \5   6 /      \10   3 /

one of the solutions of our inequality is:
x<−4π3x < - \frac{4 \pi}{3}
 _____           _____           _____           _____           _____          
      \         /     \         /     \         /     \         /
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
       x2      x3      x4      x1      x5      x6      x7      x8

Other solutions will get with the changeover to the next point
etc.
The answer:
x<−4π3x < - \frac{4 \pi}{3}
x>−π∧x<−2π3x > - \pi \wedge x < - \frac{2 \pi}{3}
x>0∧x<2π3x > 0 \wedge x < \frac{2 \pi}{3}
x>π∧x<4π3x > \pi \wedge x < \frac{4 \pi}{3}
x>2πx > 2 \pi
Solving inequality on a graph
0-80-60-40-20204060805-5
Rapid solution [src] 
  /   /           2*pi\     /            4*pi\\
Or|And|0 < x, x < ----|, And|pi < x, x < ----||
  \   \            3  /     \             3  //
(0<x∧x<2π3)∨(π<x∧x<4π3)\left(0 < x \wedge x < \frac{2 \pi}{3}\right) \vee \left(\pi < x \wedge x < \frac{4 \pi}{3}\right) 
((0 < x)∧(x < 2*pi/3))∨((pi < x)∧(x < 4*pi/3))
Rapid solution 2 [src] 
    2*pi         4*pi 
(0, ----) U (pi, ----)
     3            3
x in (0,2π3)∪(π,4π3)x\ in\ \left(0, \frac{2 \pi}{3}\right) \cup \left(\pi, \frac{4 \pi}{3}\right) 
x in Union(Interval.open(0, 2*pi/3), Interval.open(pi, 4*pi/3))
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