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Solving inequalities/ logsqrt3x<=2

logsqrt3x<=2 inequation

A inequation with variable

The solution

You have entered [src] 
   /  _____\     
log\\/ 3*x / <= 2
log⁡(3x)≤2\log{\left(\sqrt{3 x} \right)} \leq 2 
log(sqrt(3*x)) <= 2
Detail solution
Given the inequality:
log⁡(3x)≤2\log{\left(\sqrt{3 x} \right)} \leq 2
To solve this inequality, we must first solve the corresponding equation:
log⁡(3x)=2\log{\left(\sqrt{3 x} \right)} = 2
Solve:
x1=e43x_{1} = \frac{e^{4}}{3}
x1=e43x_{1} = \frac{e^{4}}{3}
This roots
x1=e43x_{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:
x0≤x1x_{0} \leq x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−110+e43- \frac{1}{10} + \frac{e^{4}}{3}
=
−110+e43- \frac{1}{10} + \frac{e^{4}}{3}
substitute to the expression
log⁡(3x)≤2\log{\left(\sqrt{3 x} \right)} \leq 2
log⁡(3(−110+e43))≤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≤e43x \leq \frac{e^{4}}{3}
 _____          
      \    
-------•-------
       x1
Solving inequality on a graph
0-30-20-10102030405060705-5
Rapid solution [src] 
   /      4       \
   |     e        |
And|x <= --, 0 < x|
   \     3        /
x≤e43∧0<xx \leq \frac{e^{4}}{3} \wedge 0 < x 
(0 < x)∧(x <= exp(4)/3)
Rapid solution 2 [src] 
     4 
    e  
(0, --]
    3
x in (0,e43]x\ in\ \left(0, \frac{e^{4}}{3}\right] 
x in Interval.Lopen(0, exp(4)/3)
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