Mister Exam
x*log(4)*x>1 inequation
The solution
Detail solution
Given the inequality:
$$x x \log{\left(4 \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$x x \log{\left(4 \right)} = 1$$
Solve:
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$x x \log{\left(4 \right)} = 1$$
to
$$x x \log{\left(4 \right)} - 1 = 0$$
Expand the expression in the equation
$$x x \log{\left(4 \right)} - 1 = 0$$
We get the quadratic equation
$$2 x^{2} \log{\left(2 \right)} - 1 = 0$$
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
$$a = 2 \log{\left(2 \right)}$$
$$b = 0$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
This roots
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} + - \frac{1}{10}$$
=
$$- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} - \frac{1}{10}$$
substitute to the expression
$$x x \log{\left(4 \right)} > 1$$
$$\left(- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} - \frac{1}{10}\right) \log{\left(4 \right)} \left(- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} - \frac{1}{10}\right) > 1$$
one of the solutions of our inequality is:
$$x < - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x > \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x x \log{\left(4 \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$x x \log{\left(4 \right)} = 1$$
Solve:
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$x x \log{\left(4 \right)} = 1$$
to
$$x x \log{\left(4 \right)} - 1 = 0$$
Expand the expression in the equation
$$x x \log{\left(4 \right)} - 1 = 0$$
We get the quadratic equation
$$2 x^{2} \log{\left(2 \right)} - 1 = 0$$
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
$$a = 2 \log{\left(2 \right)}$$
$$b = 0$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (2*log(2)) * (-1) = 8*log(2)
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
This roots
$$x_{2} = - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} + - \frac{1}{10}$$
=
$$- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} - \frac{1}{10}$$
substitute to the expression
$$x x \log{\left(4 \right)} > 1$$
$$\left(- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} - \frac{1}{10}\right) \log{\left(4 \right)} \left(- \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} - \frac{1}{10}\right) > 1$$
2
/ ___ \
| 1 \/ 2 |
|- -- - ------------| *log(4) > 1
| 10 ________|
\ 2*\/ log(2) /
one of the solutions of our inequality is:
$$x < - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
_____ _____
\ /
-------ο-------ο-------
x2 x1Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
$$x > \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}$$
Solving inequality on a graph
Rapid solution
[src]
/ / ___ \ / ___ \\ | | -\/ 2 | | \/ 2 || Or|And|-oo < x, x < ------------|, And|x < oo, ------------ < x|| | | ________| | ________ || \ \ 2*\/ log(2) / \ 2*\/ log(2) //
$$\left(-\infty < x \wedge x < - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}\right) \vee \left(x < \infty \wedge \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}} < x\right)$$
((-oo < x)∧(x < -sqrt(2)/(2*sqrt(log(2)))))∨((x < oo)∧(sqrt(2)/(2*sqrt(log(2))) < x))
Rapid solution 2
[src]
___ ___
-\/ 2 \/ 2
(-oo, ------------) U (------------, oo)
________ ________
2*\/ log(2) 2*\/ log(2)
$$x\ in\ \left(-\infty, - \frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}\right) \cup \left(\frac{\sqrt{2}}{2 \sqrt{\log{\left(2 \right)}}}, \infty\right)$$
x in Union(Interval.open(-oo, -sqrt(2)/(2*sqrt(log(2)))), Interval.open(sqrt(2)/(2*sqrt(log(2))), oo))