Mister Exam
log_(1/2)(x^2-2x-3)>4 inequation
The solution
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[src]
/ 2 \
log\x - 2*x - 3/
----------------- > 4
log(1/2)
$$\frac{\log{\left(\left(x^{2} - 2 x\right) - 3 \right)}}{\log{\left(\frac{1}{2} \right)}} > 4$$
log(x^2 - 2*x - 3)/log(1/2) > 4
Detail solution
Given the inequality:
$$\frac{\log{\left(\left(x^{2} - 2 x\right) - 3 \right)}}{\log{\left(\frac{1}{2} \right)}} > 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(\left(x^{2} - 2 x\right) - 3 \right)}}{\log{\left(\frac{1}{2} \right)}} = 4$$
Solve:
$$x_{1} = 1 - \frac{\sqrt{65}}{4}$$
$$x_{2} = 1 + \frac{\sqrt{65}}{4}$$
$$x_{1} = 1 - \frac{\sqrt{65}}{4}$$
$$x_{2} = 1 + \frac{\sqrt{65}}{4}$$
This roots
$$x_{1} = 1 - \frac{\sqrt{65}}{4}$$
$$x_{2} = 1 + \frac{\sqrt{65}}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(1 - \frac{\sqrt{65}}{4}\right) + - \frac{1}{10}$$
=
$$\frac{9}{10} - \frac{\sqrt{65}}{4}$$
substitute to the expression
$$\frac{\log{\left(\left(x^{2} - 2 x\right) - 3 \right)}}{\log{\left(\frac{1}{2} \right)}} > 4$$
$$\frac{\log{\left(-3 + \left(\left(\frac{9}{10} - \frac{\sqrt{65}}{4}\right)^{2} - 2 \left(\frac{9}{10} - \frac{\sqrt{65}}{4}\right)\right) \right)}}{\log{\left(\frac{1}{2} \right)}} > 4$$
Then
$$x < 1 - \frac{\sqrt{65}}{4}$$
no execute
one of the solutions of our inequality is:
$$x > 1 - \frac{\sqrt{65}}{4} \wedge x < 1 + \frac{\sqrt{65}}{4}$$
$$\frac{\log{\left(\left(x^{2} - 2 x\right) - 3 \right)}}{\log{\left(\frac{1}{2} \right)}} > 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(\left(x^{2} - 2 x\right) - 3 \right)}}{\log{\left(\frac{1}{2} \right)}} = 4$$
Solve:
$$x_{1} = 1 - \frac{\sqrt{65}}{4}$$
$$x_{2} = 1 + \frac{\sqrt{65}}{4}$$
$$x_{1} = 1 - \frac{\sqrt{65}}{4}$$
$$x_{2} = 1 + \frac{\sqrt{65}}{4}$$
This roots
$$x_{1} = 1 - \frac{\sqrt{65}}{4}$$
$$x_{2} = 1 + \frac{\sqrt{65}}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(1 - \frac{\sqrt{65}}{4}\right) + - \frac{1}{10}$$
=
$$\frac{9}{10} - \frac{\sqrt{65}}{4}$$
substitute to the expression
$$\frac{\log{\left(\left(x^{2} - 2 x\right) - 3 \right)}}{\log{\left(\frac{1}{2} \right)}} > 4$$
$$\frac{\log{\left(-3 + \left(\left(\frac{9}{10} - \frac{\sqrt{65}}{4}\right)^{2} - 2 \left(\frac{9}{10} - \frac{\sqrt{65}}{4}\right)\right) \right)}}{\log{\left(\frac{1}{2} \right)}} > 4$$
/ 2 \
| / ____\ ____|
| 24 |9 \/ 65 | \/ 65 |
-log|- -- + |-- - ------| + ------| > 4
\ 5 \10 4 / 2 /
-------------------------------------
log(2) Then
$$x < 1 - \frac{\sqrt{65}}{4}$$
no execute
one of the solutions of our inequality is:
$$x > 1 - \frac{\sqrt{65}}{4} \wedge x < 1 + \frac{\sqrt{65}}{4}$$
_____
/ \
-------ο-------ο-------
x1 x2
Solving inequality on a graph
Rapid solution 2
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____ ____
\/ 65 \/ 65
(1 - ------, -1) U (3, 1 + ------)
4 4
$$x\ in\ \left(1 - \frac{\sqrt{65}}{4}, -1\right) \cup \left(3, 1 + \frac{\sqrt{65}}{4}\right)$$
x in Union(Interval.open(3, 1 + sqrt(65)/4), Interval.open(1 - sqrt(65)/4, -1))
Rapid solution
[src]
/ / ____\ / ____ \\ | | \/ 65 | | \/ 65 || Or|And|3 < x, x < 1 + ------|, And|x < -1, 1 - ------ < x|| \ \ 4 / \ 4 //
$$\left(3 < x \wedge x < 1 + \frac{\sqrt{65}}{4}\right) \vee \left(x < -1 \wedge 1 - \frac{\sqrt{65}}{4} < x\right)$$
((3 < x)∧(x < 1 + sqrt(65)/4))∨((x < -1)∧(1 - sqrt(65)/4 < x))