Mister Exam
log⅓(x+2)≥-2 inequation
The solution
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log(1/3)*(x + 2) >= -2
$$\left(x + 2\right) \log{\left(\frac{1}{3} \right)} \geq -2$$
(x + 2)*log(1/3) >= -2
Detail solution
Given the inequality:
$$\left(x + 2\right) \log{\left(\frac{1}{3} \right)} \geq -2$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x + 2\right) \log{\left(\frac{1}{3} \right)} = -2$$
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:
$$- x \log{\left(3 \right)} - 2 \log{\left(3 \right)} = -2$$
Divide both parts of the equation by (-2*log(3) - x*log(3))/x
We get the answer: x = -2 + 2/log(3)
$$x_{1} = -2 + \frac{2}{\log{\left(3 \right)}}$$
$$x_{1} = -2 + \frac{2}{\log{\left(3 \right)}}$$
This roots
$$x_{1} = -2 + \frac{2}{\log{\left(3 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(-2 + \frac{2}{\log{\left(3 \right)}}\right) + - \frac{1}{10}$$
=
$$- \frac{21}{10} + \frac{2}{\log{\left(3 \right)}}$$
substitute to the expression
$$\left(x + 2\right) \log{\left(\frac{1}{3} \right)} \geq -2$$
$$\left(\left(- \frac{21}{10} + \frac{2}{\log{\left(3 \right)}}\right) + 2\right) \log{\left(\frac{1}{3} \right)} \geq -2$$
the solution of our inequality is:
$$x \leq -2 + \frac{2}{\log{\left(3 \right)}}$$
$$\left(x + 2\right) \log{\left(\frac{1}{3} \right)} \geq -2$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x + 2\right) \log{\left(\frac{1}{3} \right)} = -2$$
Solve:
Given the equation:
log(1/3)*(x+2) = -2
Expand expressions:
-2*log(3) - x*log(3) = -2
Reducing, you get:
2 - 2*log(3) - x*log(3) = 0
Expand brackets in the left part
2 - 2*log3 - x*log3 = 0
Move free summands (without x)
from left part to right part, we given:
$$- x \log{\left(3 \right)} - 2 \log{\left(3 \right)} = -2$$
Divide both parts of the equation by (-2*log(3) - x*log(3))/x
x = -2 / ((-2*log(3) - x*log(3))/x)
We get the answer: x = -2 + 2/log(3)
$$x_{1} = -2 + \frac{2}{\log{\left(3 \right)}}$$
$$x_{1} = -2 + \frac{2}{\log{\left(3 \right)}}$$
This roots
$$x_{1} = -2 + \frac{2}{\log{\left(3 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(-2 + \frac{2}{\log{\left(3 \right)}}\right) + - \frac{1}{10}$$
=
$$- \frac{21}{10} + \frac{2}{\log{\left(3 \right)}}$$
substitute to the expression
$$\left(x + 2\right) \log{\left(\frac{1}{3} \right)} \geq -2$$
$$\left(\left(- \frac{21}{10} + \frac{2}{\log{\left(3 \right)}}\right) + 2\right) \log{\left(\frac{1}{3} \right)} \geq -2$$
/ 1 2 \ -|- -- + ------|*log(3) >= -2 \ 10 log(3)/
the solution of our inequality is:
$$x \leq -2 + \frac{2}{\log{\left(3 \right)}}$$
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Solving inequality on a graph
Rapid solution 2
[src]
2*(1 - log(3))
(-oo, --------------]
log(3)
$$x\ in\ \left(-\infty, \frac{2 \left(1 - \log{\left(3 \right)}\right)}{\log{\left(3 \right)}}\right]$$
x in Interval(-oo, 2*(1 - log(3))/log(3))
Rapid solution
[src]
/ 2*(1 - log(3)) \ And|x <= --------------, -oo < x| \ log(3) /
$$x \leq \frac{2 \left(1 - \log{\left(3 \right)}\right)}{\log{\left(3 \right)}} \wedge -\infty < x$$
(-oo < x)∧(x <= 2*(1 - log(3))/log(3))