x^2log343(5-x)

+ inequation

Solve the inequality!

A inequation with variable

The solution

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 2                 / 2            \
x *log(5 - x)   log\x  - 10*x + 25/
------------- < -------------------
   log(343)            log(7)      

$$\frac{x^{2} \log{\left(- x + 5 \right)}}{\log{\left(343 \right)}} < \frac{\log{\left(x^{2} - 10 x + 25 \right)}}{\log{\left(7 \right)}}$$

x^2*log(5 - x)/log(343) < log(x^2 - 10*x + 25)/log(7)

Detail solution

Given the inequality:
$$\frac{x^{2} \log{\left(- x + 5 \right)}}{\log{\left(343 \right)}} < \frac{\log{\left(x^{2} - 10 x + 25 \right)}}{\log{\left(7 \right)}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x^{2} \log{\left(- x + 5 \right)}}{\log{\left(343 \right)}} = \frac{\log{\left(x^{2} - 10 x + 25 \right)}}{\log{\left(7 \right)}}$$
Solve:
$$x_{1} = 2.44948974278318$$
$$x_{2} = 4$$
$$x_{3} = -2.44948974278318$$
$$x_{1} = 2.44948974278318$$
$$x_{2} = 4$$
$$x_{3} = -2.44948974278318$$
This roots
$$x_{3} = -2.44948974278318$$
$$x_{1} = 2.44948974278318$$
$$x_{2} = 4$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{3}$$
For example, let's take the point
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$-2.44948974278318 - \frac{1}{10}$$
=
$$-2.54948974278318$$
substitute to the expression
$$\frac{x^{2} \log{\left(- x + 5 \right)}}{\log{\left(343 \right)}} < \frac{\log{\left(x^{2} - 10 x + 25 \right)}}{\log{\left(7 \right)}}$$
$$\frac{\left(-2.54948974278318\right)^{2} \log{\left(\left(-1\right) \left(-2.54948974278318\right) + 5 \right)}}{\log{\left(343 \right)}} < \frac{\log{\left(\left(-2.54948974278318\right)^{2} + 25 - 10 \left(-2.54948974278318\right) \right)}}{\log{\left(7 \right)}}$$

13.1394135571088   4.04295995447944
---------------- < ----------------
    log(343)            log(7)     

but

13.1394135571088   4.04295995447944
---------------- > ----------------
    log(343)            log(7)     

Then
$$x < -2.44948974278318$$
no execute
one of the solutions of our inequality is:
$$x > -2.44948974278318 \wedge x < 2.44948974278318$$

         _____           _____  
        /     \         /
-------ο-------ο-------ο-------
       x_3      x_1      x_2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > -2.44948974278318 \wedge x < 2.44948974278318$$
$$x > 4$$

Solving inequality on a graph

The graph