Mister Exam
-2*sin(x/2)*cos(x/2)>0 inequation
The solution
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-2*sin|-|*cos|-| > 0
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$$- 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} > 0$$
(-2*sin(x/2))*cos(x/2) > 0
Detail solution
Given the inequality:
$$- 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = - \pi$$
$$x_{3} = \pi$$
$$x_{1} = 0$$
$$x_{2} = - \pi$$
$$x_{3} = \pi$$
This roots
$$x_{2} = - \pi$$
$$x_{1} = 0$$
$$x_{3} = \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \pi - \frac{1}{10}$$
=
$$- \pi - \frac{1}{10}$$
substitute to the expression
$$- 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} > 0$$
$$- 2 \sin{\left(\frac{- \pi - \frac{1}{10}}{2} \right)} \cos{\left(\frac{- \pi - \frac{1}{10}}{2} \right)} > 0$$
Then
$$x < - \pi$$
no execute
one of the solutions of our inequality is:
$$x > - \pi \wedge x < 0$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > - \pi \wedge x < 0$$
$$x > \pi$$
$$- 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = - \pi$$
$$x_{3} = \pi$$
$$x_{1} = 0$$
$$x_{2} = - \pi$$
$$x_{3} = \pi$$
This roots
$$x_{2} = - \pi$$
$$x_{1} = 0$$
$$x_{3} = \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \pi - \frac{1}{10}$$
=
$$- \pi - \frac{1}{10}$$
substitute to the expression
$$- 2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} > 0$$
$$- 2 \sin{\left(\frac{- \pi - \frac{1}{10}}{2} \right)} \cos{\left(\frac{- \pi - \frac{1}{10}}{2} \right)} > 0$$
-2*cos(1/20)*sin(1/20) > 0
Then
$$x < - \pi$$
no execute
one of the solutions of our inequality is:
$$x > - \pi \wedge x < 0$$
_____ _____
/ \ /
-------ο-------ο-------ο-------
x2 x1 x3Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > - \pi \wedge x < 0$$
$$x > \pi$$
Solving inequality on a graph