Mister Exam

# tgx>=-1 inequation

A inequation with variable

### The solution

You have entered [src]
tan(x) >= -1
$$\tan{\left(x \right)} \geq -1$$
tan(x) >= -1
Detail solution
Given the inequality:
$$\tan{\left(x \right)} \geq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} = -1$$
Solve:
Given the equation
$$\tan{\left(x \right)} = -1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(-1 \right)}$$
Or
$$x = \pi n - \frac{\pi}{4}$$
, where n - is a integer
$$x_{1} = \pi n - \frac{\pi}{4}$$
$$x_{1} = \pi n - \frac{\pi}{4}$$
This roots
$$x_{1} = \pi n - \frac{\pi}{4}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{4}\right) - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{4} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x \right)} \geq -1$$
$$\tan{\left(\pi n - \frac{\pi}{4} - \frac{1}{10} \right)} \geq -1$$
    /1    pi\
-tan|-- + --| >= -1
\10   4 /      

but
    /1    pi\
-tan|-- + --| < -1
\10   4 /     

Then
$$x \leq \pi n - \frac{\pi}{4}$$
no execute
the solution of our inequality is:
$$x \geq \pi n - \frac{\pi}{4}$$
         _____
/
-------•-------
x_1
Solving inequality on a graph
Rapid solution 2 [src]
    pi     3*pi
[0, --) U [----, pi)
2       4       
$$x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left[\frac{3 \pi}{4}, \pi\right)$$
x in Union(Interval.Ropen(0, pi/2), Interval.Ropen(3*pi/4, pi))
Rapid solution [src]
  /   /            pi\     /3*pi             \\
Or|And|0 <= x, x < --|, And|---- <= x, x < pi||
\   \            2 /     \ 4               //
$$\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(\frac{3 \pi}{4} \leq x \wedge x < \pi\right)$$
((0 <= x)∧(x < pi/2))∨((x < pi)∧(3*pi/4 <= x))
The graph