Mister Exam
logsqrt3x<=2 inequation
The solution
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/ _____\ log\\/ 3*x / <= 2
$$\log{\left(\sqrt{3 x} \right)} \leq 2$$
log(sqrt(3*x)) <= 2
Detail solution
Given the inequality:
$$\log{\left(\sqrt{3 x} \right)} \leq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\sqrt{3 x} \right)} = 2$$
Solve:
$$x_{1} = \frac{e^{4}}{3}$$
$$x_{1} = \frac{e^{4}}{3}$$
This roots
$$x_{1} = \frac{e^{4}}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{e^{4}}{3}$$
=
$$- \frac{1}{10} + \frac{e^{4}}{3}$$
substitute to the expression
$$\log{\left(\sqrt{3 x} \right)} \leq 2$$
$$\log{\left(\sqrt{3 \left(- \frac{1}{10} + \frac{e^{4}}{3}\right)} \right)} \leq 2$$
the solution of our inequality is:
$$x \leq \frac{e^{4}}{3}$$
$$\log{\left(\sqrt{3 x} \right)} \leq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\sqrt{3 x} \right)} = 2$$
Solve:
$$x_{1} = \frac{e^{4}}{3}$$
$$x_{1} = \frac{e^{4}}{3}$$
This roots
$$x_{1} = \frac{e^{4}}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{e^{4}}{3}$$
=
$$- \frac{1}{10} + \frac{e^{4}}{3}$$
substitute to the expression
$$\log{\left(\sqrt{3 x} \right)} \leq 2$$
$$\log{\left(\sqrt{3 \left(- \frac{1}{10} + \frac{e^{4}}{3}\right)} \right)} \leq 2$$
/ ___________\
| / 3 4 |
log| / - -- + e | <= 2
\\/ 10 /
the solution of our inequality is:
$$x \leq \frac{e^{4}}{3}$$
_____
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x1
Solving inequality on a graph
Rapid solution
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/ 4 \ | e | And|x <= --, 0 < x| \ 3 /
$$x \leq \frac{e^{4}}{3} \wedge 0 < x$$
(0 < x)∧(x <= exp(4)/3)
Rapid solution 2
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4
e
(0, --]
3
$$x\ in\ \left(0, \frac{e^{4}}{3}\right]$$
x in Interval.Lopen(0, exp(4)/3)