Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- -2cos(x/3)<sqrt(3)
- tan(x+pi/6)>sqrt(3)/3
- sin(6x)<sqrt-3/2
- logx(3)<0
- Derivative of:
-
sqrt(3)
- Integral of d{x}:
-
sqrt(3)
- The double integral of:
- sqrt(3)
-
Identical expressions
- -2cos(x/ three)<sqrt(three)
- minus 2 co sinus of e of (x divide by 3) less than square root of (3)
- minus 2 co sinus of e of (x divide by three) less than square root of (three)
- -2cos(x/3)<√(3)
- -2cosx/3<sqrt3
- -2cos(x divide by 3)<sqrt(3)
-
Similar expressions
- 2cos(x/3)<sqrt(3)
- -2cosx/3<
- -2cosx/3<cbrt3
- -2*cos(x/3)>1
- -2cosx/3<3
- sqrt(3^(x-54))-7*sqrt(3^(x-58))<162
-2cos(x/3)
The solution
You have entered
[src]
/x\ ___
-2*cos|-| < \/ 3
\3/
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
-2*cos(x/3) < sqrt(3)
Detail solution
Given the inequality:
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \cos{\left(\frac{x}{3} \right)} = \sqrt{3}$$
Solve:
Given the equation
$$- 2 \cos{\left(\frac{x}{3} \right)} = \sqrt{3}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -2
The equation is transformed to
$$\cos{\left(\frac{x}{3} \right)} = - \frac{\sqrt{3}}{2}$$
This equation is transformed to
$$\frac{x}{3} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$\frac{x}{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
Or
$$\frac{x}{3} = \pi n + \frac{5 \pi}{6}$$
$$\frac{x}{3} = \pi n - \frac{\pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{3}$$
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
This roots
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(3 \pi n + \frac{5 \pi}{2}\right) + - \frac{1}{10}$$
=
$$3 \pi n - \frac{1}{10} + \frac{5 \pi}{2}$$
substitute to the expression
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
$$- 2 \cos{\left(\frac{3 \pi n - \frac{1}{10} + \frac{5 \pi}{2}}{3} \right)} < \sqrt{3}$$
/ 1 pi \ ___
2*sin|- -- + -- + pi*n| < \/ 3
\ 30 3 /
one of the solutions of our inequality is:
$$x < 3 \pi n + \frac{5 \pi}{2}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 3 \pi n + \frac{5 \pi}{2}$$
$$x > 3 \pi n - \frac{\pi}{2}$$
Solving inequality on a graph
Rapid solution 2
[src]
/ _____________\ / _____________\
| / ___ | | / ___ |
[0, 6*atan\\/ 7 + 4*\/ 3 /) U (- 6*atan\\/ 7 + 4*\/ 3 / + 6*pi, 6*pi]
$$x\ in\ \left[0, 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}\right) \cup \left(- 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} + 6 \pi, 6 \pi\right]$$
x in Union(Interval.Ropen(0, 6*atan(sqrt(4*sqrt(3) + 7))), Interval.Lopen(-6*atan(sqrt(4*sqrt(3) + 7)) + 6*pi, 6*pi))
Rapid solution
[src]
/ / / _____________\\ / / _____________\ \\
| | | / ___ || | | / ___ | ||
Or\And\0 <= x, x < 6*atan\\/ 7 + 4*\/ 3 //, And\x <= 6*pi, - 6*atan\\/ 7 + 4*\/ 3 / + 6*pi < x//
$$\left(0 \leq x \wedge x < 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}\right) \vee \left(x \leq 6 \pi \wedge - 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} + 6 \pi < x\right)$$
((0 <= x)∧(x < 6*atan(sqrt(7 + 4*sqrt(3)))))∨((x <= 6*pi)∧(-6*atan(sqrt(7 + 4*sqrt(3))) + 6*pi < x))
The solution
You have entered
[src]
/x\ ___
-2*cos|-| < \/ 3
\3/
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
-2*cos(x/3) < sqrt(3)
Detail solution
Given the inequality:
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \cos{\left(\frac{x}{3} \right)} = \sqrt{3}$$
Solve:
Given the equation
$$- 2 \cos{\left(\frac{x}{3} \right)} = \sqrt{3}$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\cos{\left(\frac{x}{3} \right)} = - \frac{\sqrt{3}}{2}$$
This equation is transformed to
$$\frac{x}{3} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$\frac{x}{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
Or
$$\frac{x}{3} = \pi n + \frac{5 \pi}{6}$$
$$\frac{x}{3} = \pi n - \frac{\pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{3}$$
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
This roots
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(3 \pi n + \frac{5 \pi}{2}\right) + - \frac{1}{10}$$
=
$$3 \pi n - \frac{1}{10} + \frac{5 \pi}{2}$$
substitute to the expression
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
$$- 2 \cos{\left(\frac{3 \pi n - \frac{1}{10} + \frac{5 \pi}{2}}{3} \right)} < \sqrt{3}$$
one of the solutions of our inequality is:
$$x < 3 \pi n + \frac{5 \pi}{2}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 3 \pi n + \frac{5 \pi}{2}$$
$$x > 3 \pi n - \frac{\pi}{2}$$
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \cos{\left(\frac{x}{3} \right)} = \sqrt{3}$$
Solve:
Given the equation
$$- 2 \cos{\left(\frac{x}{3} \right)} = \sqrt{3}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -2
The equation is transformed to
$$\cos{\left(\frac{x}{3} \right)} = - \frac{\sqrt{3}}{2}$$
This equation is transformed to
$$\frac{x}{3} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$\frac{x}{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
Or
$$\frac{x}{3} = \pi n + \frac{5 \pi}{6}$$
$$\frac{x}{3} = \pi n - \frac{\pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{3}$$
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
This roots
$$x_{1} = 3 \pi n + \frac{5 \pi}{2}$$
$$x_{2} = 3 \pi n - \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(3 \pi n + \frac{5 \pi}{2}\right) + - \frac{1}{10}$$
=
$$3 \pi n - \frac{1}{10} + \frac{5 \pi}{2}$$
substitute to the expression
$$- 2 \cos{\left(\frac{x}{3} \right)} < \sqrt{3}$$
$$- 2 \cos{\left(\frac{3 \pi n - \frac{1}{10} + \frac{5 \pi}{2}}{3} \right)} < \sqrt{3}$$
/ 1 pi \ ___
2*sin|- -- + -- + pi*n| < \/ 3
\ 30 3 / one of the solutions of our inequality is:
$$x < 3 \pi n + \frac{5 \pi}{2}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 3 \pi n + \frac{5 \pi}{2}$$
$$x > 3 \pi n - \frac{\pi}{2}$$
Solving inequality on a graph
Rapid solution 2
[src]
/ _____________\ / _____________\
| / ___ | | / ___ |
[0, 6*atan\\/ 7 + 4*\/ 3 /) U (- 6*atan\\/ 7 + 4*\/ 3 / + 6*pi, 6*pi]
$$x\ in\ \left[0, 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}\right) \cup \left(- 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} + 6 \pi, 6 \pi\right]$$
x in Union(Interval.Ropen(0, 6*atan(sqrt(4*sqrt(3) + 7))), Interval.Lopen(-6*atan(sqrt(4*sqrt(3) + 7)) + 6*pi, 6*pi))
Rapid solution
[src]
/ / / _____________\\ / / _____________\ \\ | | | / ___ || | | / ___ | || Or\And\0 <= x, x < 6*atan\\/ 7 + 4*\/ 3 //, And\x <= 6*pi, - 6*atan\\/ 7 + 4*\/ 3 / + 6*pi < x//
$$\left(0 \leq x \wedge x < 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}\right) \vee \left(x \leq 6 \pi \wedge - 6 \operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)} + 6 \pi < x\right)$$
((0 <= x)∧(x < 6*atan(sqrt(7 + 4*sqrt(3)))))∨((x <= 6*pi)∧(-6*atan(sqrt(7 + 4*sqrt(3))) + 6*pi < x))