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Other calculators
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How to use it?
- Inequation:
- (x-5)*(sqrt(9-x))>0
- 3^(log(x-1/x+2))>1/9
- 4^(x^2-x-6)-1>0
- 2^(x^2)*13^(x-1)>=2
- Integral of d{x}:
- 1/9
- Limit of the function:
-
1/9
- 1/9
-
Identical expressions
- three ^(log(x- one /x+ two))> one / nine
- 3 to the power of ( logarithm of (x minus 1 divide by x plus 2)) greater than 1 divide by 9
- three to the power of ( logarithm of (x minus one divide by x plus two)) greater than one divide by nine
- 3(log(x-1/x+2))>1/9
- 3logx-1/x+2>1/9
- 3^logx-1/x+2>1/9
- 3^(log(x-1 divide by x+2))>1 divide by 9
-
Similar expressions
- 3^(log(x-1/x-2))>1/9
- 3^(log(x+1/x+2))>1/9
3^(log(x-1/x+2))>1/9 inequation
The solution
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/ 1 \
log|x - - + 2|
\ x /
3 > 1/9
$$3^{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}} > \frac{1}{9}$$
3^log(x - 1/x + 2) > 1/9
Detail solution
Given the inequality:
$$3^{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}} > \frac{1}{9}$$
To solve this inequality, we must first solve the corresponding equation:
$$3^{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}} = \frac{1}{9}$$
Solve:
$$x_{1} = \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
$$x_{2} = \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
$$x_{1} = \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
$$x_{2} = \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
This roots
$$x_{1} = \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
$$x_{2} = \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 \left(e^{1}\right)^{2}} + - \frac{1}{10}$$
=
$$\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}} - \frac{1}{10}$$
substitute to the expression
$$3^{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}} > \frac{1}{9}$$
$$3^{\log{\left(\left(\left(\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 \left(e^{1}\right)^{2}} - \frac{1}{10}\right) - \frac{1}{\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 \left(e^{1}\right)^{2}} - \frac{1}{10}}\right) + 2 \right)}} > \frac{1}{9}$$
Then
$$x < \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}} \wedge x < \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
$$3^{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}} > \frac{1}{9}$$
To solve this inequality, we must first solve the corresponding equation:
$$3^{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}} = \frac{1}{9}$$
Solve:
$$x_{1} = \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
$$x_{2} = \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
$$x_{1} = \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
$$x_{2} = \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
This roots
$$x_{1} = \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
$$x_{2} = \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 \left(e^{1}\right)^{2}} + - \frac{1}{10}$$
=
$$\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}} - \frac{1}{10}$$
substitute to the expression
$$3^{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}} > \frac{1}{9}$$
$$3^{\log{\left(\left(\left(\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 \left(e^{1}\right)^{2}} - \frac{1}{10}\right) - \frac{1}{\frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 \left(e^{1}\right)^{2}} - \frac{1}{10}}\right) + 2 \right)}} > \frac{1}{9}$$
/ / _________________ \ \
| | / 2 4 2| -2|
|19 1 \1 - \/ 1 - 4*e + 8*e - 2*e /*e |
log|-- - -------------------------------------------- + -------------------------------------|
|10 / _________________ \ 2 |
| | / 2 4 2| -2 | > 1/9
| 1 \1 - \/ 1 - 4*e + 8*e - 2*e /*e |
| - -- + ------------------------------------- |
\ 10 2 /
3
Then
$$x < \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{- \sqrt{- 4 e^{2} + 1 + 8 e^{4}} - 2 e^{2} + 1}{2 e^{2}} \wedge x < \frac{- 2 e^{2} + 1 + \sqrt{- 4 e^{2} + 1 + 8 e^{4}}}{2 e^{2}}$$
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/ \
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x1 x2
Solving inequality on a graph