Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- x^2>-x-2
- (1/2)*x-4*(x-3)>3*x
-
sin2x>-sqrt3/2
- cos(9*x)>=sqrt(2)/2
-
Identical expressions
- tan2(x)-4tan(x)+ three > zero
- tangent of 2(x) minus 4 tangent of (x) plus 3 greater than 0
- tangent of 2(x) minus 4 tangent of (x) plus three greater than zero
- tan2x-4tanx+3>0
-
Similar expressions
- tan2(x)-4tan(x)-3>0
- tan2(x)+4tan(x)+3>0
tan2(x)-4tan(x)+3>0 inequation
The solution
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2 tan (x) - 4*tan(x) + 3 > 0
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 > 0$$
tan(x)^2 - 4*tan(x) + 3 > 0
Detail solution
Given the inequality:
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 = 0$$
Solve:
Given the equation
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 = 0$$
transform
$$\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 3 = 0$$
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -4$$
$$c = 3$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 3$$
$$w_{2} = 1$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(3 \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(3 \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(1 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{4}$$
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(3 \right)}$$
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(3 \right)}$$
This roots
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(3 \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{4}$$
=
$$- \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 > 0$$
$$\left(- 4 \tan{\left(- \frac{1}{10} + \frac{\pi}{4} \right)} + \tan^{2}{\left(- \frac{1}{10} + \frac{\pi}{4} \right)}\right) + 3 > 0$$
one of the solutions of our inequality is:
$$x < \frac{\pi}{4}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{\pi}{4}$$
$$x > \operatorname{atan}{\left(3 \right)}$$
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 = 0$$
Solve:
Given the equation
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 = 0$$
transform
$$\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 3 = 0$$
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -4$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (1) * (3) = 4
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 3$$
$$w_{2} = 1$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(3 \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(3 \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(1 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{4}$$
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(3 \right)}$$
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(3 \right)}$$
This roots
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(3 \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{4}$$
=
$$- \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\left(\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}\right) + 3 > 0$$
$$\left(- 4 \tan{\left(- \frac{1}{10} + \frac{\pi}{4} \right)} + \tan^{2}{\left(- \frac{1}{10} + \frac{\pi}{4} \right)}\right) + 3 > 0$$
2/1 pi\ /1 pi\
3 + cot |-- + --| - 4*cot|-- + --| > 0
\10 4 / \10 4 / one of the solutions of our inequality is:
$$x < \frac{\pi}{4}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{\pi}{4}$$
$$x > \operatorname{atan}{\left(3 \right)}$$
Solving inequality on a graph
Rapid solution 2
[src]
pi pi pi
[0, --) U (atan(3), --) U (--, pi]
4 2 2
$$x\ in\ \left[0, \frac{\pi}{4}\right) \cup \left(\operatorname{atan}{\left(3 \right)}, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$$
x in Union(Interval.Ropen(0, pi/4), Interval.Lopen(pi/2, pi), Interval.open(atan(3), pi/2))
Rapid solution
[src]
/ / pi\ / pi \ / pi \\ Or|And|0 <= x, x < --|, And|x <= pi, -- < x|, And|x < --, atan(3) < x|| \ \ 4 / \ 2 / \ 2 //
$$\left(0 \leq x \wedge x < \frac{\pi}{4}\right) \vee \left(x \leq \pi \wedge \frac{\pi}{2} < x\right) \vee \left(x < \frac{\pi}{2} \wedge \operatorname{atan}{\left(3 \right)} < x\right)$$
((0 <= x)∧(x < pi/4))∨((x <= pi)∧(pi/2 < x))∨((atan(3) < x)∧(x < pi/2))