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Solving inequalities/ 1/5lgx+2/1+lgx<1

1/5lgx+2/1+lgx<1 inequation

A inequation with variable

The solution

You have entered [src] 
log(x)                 
------ + 2 + log(x) < 1
  5
(log⁡(x)5+2)+log⁡(x)<1\left(\frac{\log{\left(x \right)}}{5} + 2\right) + \log{\left(x \right)} < 1 
log(x)/5 + 2 + log(x) < 1
Detail solution
Given the inequality:
(log⁡(x)5+2)+log⁡(x)<1\left(\frac{\log{\left(x \right)}}{5} + 2\right) + \log{\left(x \right)} < 1
To solve this inequality, we must first solve the corresponding equation:
(log⁡(x)5+2)+log⁡(x)=1\left(\frac{\log{\left(x \right)}}{5} + 2\right) + \log{\left(x \right)} = 1
Solve:
Given the equation
(log⁡(x)5+2)+log⁡(x)=1\left(\frac{\log{\left(x \right)}}{5} + 2\right) + \log{\left(x \right)} = 1
6log⁡(x)5=−1\frac{6 \log{\left(x \right)}}{5} = -1
Let's divide both parts of the equation by the multiplier of log =6/5
log⁡(x)=−56\log{\left(x \right)} = - \frac{5}{6}
This equation is of the form:
log(v)=p

By definition log
v=e^p

then
x=e−165x = e^{- \frac{1}{\frac{6}{5}}}
simplify
x=e−56x = e^{- \frac{5}{6}}
x1=e−56x_{1} = e^{- \frac{5}{6}}
x1=e−56x_{1} = e^{- \frac{5}{6}}
This roots
x1=e−56x_{1} = e^{- \frac{5}{6}}
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}
=
−110+e−56- \frac{1}{10} + e^{- \frac{5}{6}}
=
−110+e−56- \frac{1}{10} + e^{- \frac{5}{6}}
substitute to the expression
(log⁡(x)5+2)+log⁡(x)<1\left(\frac{\log{\left(x \right)}}{5} + 2\right) + \log{\left(x \right)} < 1
log⁡(−110+e−56)+(log⁡(−110+e−56)5+2)<1\log{\left(- \frac{1}{10} + e^{- \frac{5}{6}} \right)} + \left(\frac{\log{\left(- \frac{1}{10} + e^{- \frac{5}{6}} \right)}}{5} + 2\right) < 1
         /  1     -5/6\    
    6*log|- -- + e    |    
         \  10        / < 1
2 + -------------------    
             5

the solution of our inequality is:
x<e−56x < e^{- \frac{5}{6}}
 _____          
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       x1
Solving inequality on a graph
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-1010
Rapid solution [src] 
   /            -5/6\
And\0 < x, x < e    /
0<x∧x<e−560 < x \wedge x < e^{- \frac{5}{6}} 
(0 < x)∧(x < exp(-5/6))
Rapid solution 2 [src] 
     -5/6 
(0, e    )
x in (0,e−56)x\ in\ \left(0, e^{- \frac{5}{6}}\right) 
x in Interval.open(0, exp(-5/6))
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