Mister Exam
log(1/2)(1-x)>=-3 inequation
The solution
You have entered
[src]
log(1/2)*(1 - x) >= -3
$$\left(1 - x\right) \log{\left(\frac{1}{2} \right)} \geq -3$$
(1 - x)*log(1/2) >= -3
Detail solution
Given the inequality:
$$\left(1 - x\right) \log{\left(\frac{1}{2} \right)} \geq -3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(1 - x\right) \log{\left(\frac{1}{2} \right)} = -3$$
Solve:
Given the equation:
Expand expressions:
Reducing, you get:
Expand brackets in the left part
Move free summands (without x)
from left part to right part, we given:
$$x \log{\left(2 \right)} - \log{\left(2 \right)} = -3$$
Divide both parts of the equation by (-log(2) + x*log(2))/x
We get the answer: x = 1 - 3/log(2)
$$x_{1} = 1 - \frac{3}{\log{\left(2 \right)}}$$
$$x_{1} = 1 - \frac{3}{\log{\left(2 \right)}}$$
This roots
$$x_{1} = 1 - \frac{3}{\log{\left(2 \right)}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(1 - \frac{3}{\log{\left(2 \right)}}\right) + - \frac{1}{10}$$
=
$$\frac{9}{10} - \frac{3}{\log{\left(2 \right)}}$$
substitute to the expression
$$\left(1 - x\right) \log{\left(\frac{1}{2} \right)} \geq -3$$
$$\left(1 - \left(\frac{9}{10} - \frac{3}{\log{\left(2 \right)}}\right)\right) \log{\left(\frac{1}{2} \right)} \geq -3$$
but
Then
$$x \leq 1 - \frac{3}{\log{\left(2 \right)}}$$
no execute
the solution of our inequality is:
$$x \geq 1 - \frac{3}{\log{\left(2 \right)}}$$
$$\left(1 - x\right) \log{\left(\frac{1}{2} \right)} \geq -3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(1 - x\right) \log{\left(\frac{1}{2} \right)} = -3$$
Solve:
Given the equation:
log(1/2)*(1-x) = -3
Expand expressions:
-log(2) + x*log(2) = -3
Reducing, you get:
3 - log(2) + x*log(2) = 0
Expand brackets in the left part
3 - log2 + x*log2 = 0
Move free summands (without x)
from left part to right part, we given:
$$x \log{\left(2 \right)} - \log{\left(2 \right)} = -3$$
Divide both parts of the equation by (-log(2) + x*log(2))/x
x = -3 / ((-log(2) + x*log(2))/x)
We get the answer: x = 1 - 3/log(2)
$$x_{1} = 1 - \frac{3}{\log{\left(2 \right)}}$$
$$x_{1} = 1 - \frac{3}{\log{\left(2 \right)}}$$
This roots
$$x_{1} = 1 - \frac{3}{\log{\left(2 \right)}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(1 - \frac{3}{\log{\left(2 \right)}}\right) + - \frac{1}{10}$$
=
$$\frac{9}{10} - \frac{3}{\log{\left(2 \right)}}$$
substitute to the expression
$$\left(1 - x\right) \log{\left(\frac{1}{2} \right)} \geq -3$$
$$\left(1 - \left(\frac{9}{10} - \frac{3}{\log{\left(2 \right)}}\right)\right) \log{\left(\frac{1}{2} \right)} \geq -3$$
/1 3 \ -|-- + ------|*log(2) >= -3 \10 log(2)/
but
/1 3 \ -|-- + ------|*log(2) < -3 \10 log(2)/
Then
$$x \leq 1 - \frac{3}{\log{\left(2 \right)}}$$
no execute
the solution of our inequality is:
$$x \geq 1 - \frac{3}{\log{\left(2 \right)}}$$
_____
/
-------•-------
x1
Solving inequality on a graph
Rapid solution
[src]
/-(3 - log(2)) \ And|-------------- <= x, x < oo| \ log(2) /
$$- \frac{3 - \log{\left(2 \right)}}{\log{\left(2 \right)}} \leq x \wedge x < \infty$$
(x < oo)∧(-(3 - log(2))/log(2) <= x)
Rapid solution 2
[src]
-(3 - log(2))
[--------------, oo)
log(2)
$$x\ in\ \left[- \frac{3 - \log{\left(2 \right)}}{\log{\left(2 \right)}}, \infty\right)$$
x in Interval(-(3 - log(2))/log(2), oo)