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How to use it?
- Inequation:
- lg^2x+lgx<=2
- 15-5x>=0
-
x^2-1
- log(4*x+1)(5*x-6)+log(5*x-6)(4*x+1)<=2
-
Identical expressions
- (log^ two (one / five)*x^ two)-3 one log(1/ five)*x- eight < zero
- ( logarithm of squared (1 divide by 5) multiply by x squared ) minus 31 logarithm of (1 divide by 5) multiply by x minus 8 less than 0
- ( logarithm of to the power of two (one divide by five) multiply by x to the power of two) minus 3 one logarithm of (1 divide by five) multiply by x minus eight less than zero
- (log2(1/5)*x2)-31log(1/5)*x-8<0
- log21/5*x2-31log1/5*x-8<0
- (log²(1/5)*x²)-31log(1/5)*x-8<0
- (log to the power of 2(1/5)*x to the power of 2)-31log(1/5)*x-8<0
- (log^2(1/5)x^2)-31log(1/5)x-8<0
- (log2(1/5)x2)-31log(1/5)x-8<0
- log21/5x2-31log1/5x-8<0
- log^21/5x^2-31log1/5x-8<0
- (log^2(1 divide by 5)*x^2)-31log(1 divide by 5)*x-8<0
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Similar expressions
- (log^2(1/5)*x^2)-31log(1/5)*x+8<0
- (log^2(1/5)*x^2)+31log(1/5)*x-8<0
(log^2(1/5)*x^2)-31log(1/5)*x-8<0 inequation
The solution
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[src]
2 2 log (1/5)*x - 31*log(1/5)*x - 8 < 0
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 < 0$$
x^2*log(1/5)^2 - x*31*log(1/5) - 8 < 0
Detail solution
Given the inequality:
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 = 0$$
Solve:
Expand the expression in the equation
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 = 0$$
We get the quadratic equation
$$x^{2} \log{\left(5 \right)}^{2} + 31 x \log{\left(5 \right)} - 8 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = \log{\left(5 \right)}^{2}$$
$$b = 31 \log{\left(5 \right)}$$
$$c = -8$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
This roots
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} + - \frac{1}{10}$$
=
$$\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} - \frac{1}{10}$$
substitute to the expression
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 < 0$$
$$-8 + \left(- \left(\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} - \frac{1}{10}\right) 31 \log{\left(\frac{1}{5} \right)} + \left(\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} - \frac{1}{10}\right)^{2} \log{\left(\frac{1}{5} \right)}^{2}\right) < 0$$
but
Then
$$x < \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} \wedge x < \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 = 0$$
Solve:
Expand the expression in the equation
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 = 0$$
We get the quadratic equation
$$x^{2} \log{\left(5 \right)}^{2} + 31 x \log{\left(5 \right)} - 8 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = \log{\left(5 \right)}^{2}$$
$$b = 31 \log{\left(5 \right)}$$
$$c = -8$$
, then
D = b^2 - 4 * a * c =
(31*log(5))^2 - 4 * (log(5)^2) * (-8) = 993*log(5)^2
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
This roots
$$x_{2} = \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
$$x_{1} = \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} + - \frac{1}{10}$$
=
$$\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} - \frac{1}{10}$$
substitute to the expression
$$\left(x^{2} \log{\left(\frac{1}{5} \right)}^{2} - x 31 \log{\left(\frac{1}{5} \right)}\right) - 8 < 0$$
$$-8 + \left(- \left(\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} - \frac{1}{10}\right) 31 \log{\left(\frac{1}{5} \right)} + \left(\frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} - \frac{1}{10}\right)^{2} \log{\left(\frac{1}{5} \right)}^{2}\right) < 0$$
2
/ _____ \ / _____ \
| 1 -31*log(5) - \/ 993 *log(5)| 2 | 1 -31*log(5) - \/ 993 *log(5)|
-8 + |- -- + ---------------------------| *log (5) + 31*|- -- + ---------------------------|*log(5) < 0
| 10 2 | | 10 2 |
\ 2*log (5) / \ 2*log (5) /
but
2
/ _____ \ / _____ \
| 1 -31*log(5) - \/ 993 *log(5)| 2 | 1 -31*log(5) - \/ 993 *log(5)|
-8 + |- -- + ---------------------------| *log (5) + 31*|- -- + ---------------------------|*log(5) > 0
| 10 2 | | 10 2 |
\ 2*log (5) / \ 2*log (5) /
Then
$$x < \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{- \sqrt{993} \log{\left(5 \right)} - 31 \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}} \wedge x < \frac{- 31 \log{\left(5 \right)} + \sqrt{993} \log{\left(5 \right)}}{2 \log{\left(5 \right)}^{2}}$$
_____
/ \
-------ο-------ο-------
x2 x1
Solving inequality on a graph
Rapid solution 2
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/ _____\ / _____\
-\31 + \/ 993 / -\31 - \/ 993 /
(----------------, ----------------)
2*log(5) 2*log(5)
$$x\ in\ \left(- \frac{31 + \sqrt{993}}{2 \log{\left(5 \right)}}, - \frac{31 - \sqrt{993}}{2 \log{\left(5 \right)}}\right)$$
x in Interval.open(-(31 + sqrt(993))/(2*log(5)), -(31 - sqrt(993))/(2*log(5)))
Rapid solution
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/ _____ _____ \ | 31 \/ 993 31 \/ 993 | | - -- + ------- - -- - ------- | | 2 2 2 2 | And|x < --------------, -------------- < x| \ log(5) log(5) /
$$x < \frac{- \frac{31}{2} + \frac{\sqrt{993}}{2}}{\log{\left(5 \right)}} \wedge \frac{- \frac{\sqrt{993}}{2} - \frac{31}{2}}{\log{\left(5 \right)}} < x$$
(x < (-31/2 + sqrt(993)/2)/log(5))∧((-31/2 - sqrt(993)/2)/log(5) < x)