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Solving inequalities/ log6(x^2-5x)<2

log6(x^2-5x)<2 inequation

A inequation with variable

The solution

You have entered [src] 
   / 2      \    
log\x  - 5*x/    
------------- < 2
    log(6)
log⁡(x2−5x)log⁡(6)<2\frac{\log{\left(x^{2} - 5 x \right)}}{\log{\left(6 \right)}} < 2 
log(x^2 - 5*x)/log(6) < 2
Detail solution
Given the inequality:
log⁡(x2−5x)log⁡(6)<2\frac{\log{\left(x^{2} - 5 x \right)}}{\log{\left(6 \right)}} < 2
To solve this inequality, we must first solve the corresponding equation:
log⁡(x2−5x)log⁡(6)=2\frac{\log{\left(x^{2} - 5 x \right)}}{\log{\left(6 \right)}} = 2
Solve:
x1=−4x_{1} = -4
x2=9x_{2} = 9
x1=−4x_{1} = -4
x2=9x_{2} = 9
This roots
x1=−4x_{1} = -4
x2=9x_{2} = 9
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x1x_{0} < x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−4+−110-4 + - \frac{1}{10}
=
−4110- \frac{41}{10}
substitute to the expression
log⁡(x2−5x)log⁡(6)<2\frac{\log{\left(x^{2} - 5 x \right)}}{\log{\left(6 \right)}} < 2
log⁡((−4110)2−(−41)510)log⁡(6)<2\frac{\log{\left(\left(- \frac{41}{10}\right)^{2} - \frac{\left(-41\right) 5}{10} \right)}}{\log{\left(6 \right)}} < 2
   /3731\    
log|----|    
   \100 / < 2
---------    
  log(6)

but
   /3731\    
log|----|    
   \100 / > 2
---------    
  log(6)

Then
x<−4x < -4
no execute
one of the solutions of our inequality is:
x>−4∧x<9x > -4 \wedge x < 9
         _____  
        /     \  
-------ο-------ο-------
       x1      x2
Rapid solution 2 [src] 
(-4, 0) U (5, 9)
x in (−4,0)∪(5,9)x\ in\ \left(-4, 0\right) \cup \left(5, 9\right) 
x in Union(Interval.open(-4, 0), Interval.open(5, 9))
Rapid solution [src] 
Or(And(-4 < x, x < 0), And(5 < x, x < 9))
(−4<x∧x<0)∨(5<x∧x<9)\left(-4 < x \wedge x < 0\right) \vee \left(5 < x \wedge x < 9\right) 
((-4 < x)∧(x < 0))∨((5 < x)∧(x < 9))
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