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  • Similar expressions

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Solving inequalities/ 81^x+2*25^(x*log(5))-5/(4x-1)^2>=0

81^x+2*25^(x*log(5))-5/(4x-1)^2>=0 inequation

A inequation with variable

The solution

You have entered [src] 
  x       x*log(5)       5          
81  + 2*25         - ---------- >= 0
                              2     
                     (4*x - 1)
$$\left(2 \cdot 25^{x \log{\left(5 \right)}} + 81^{x}\right) - \frac{5}{\left(4 x - 1\right)^{2}} \geq 0$$ 
2*25^(x*log(5)) + 81^x - 5/(4*x - 1)^2 >= 0
Detail solution
Given the inequality:
$$\left(2 \cdot 25^{x \log{\left(5 \right)}} + 81^{x}\right) - \frac{5}{\left(4 x - 1\right)^{2}} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(2 \cdot 25^{x \log{\left(5 \right)}} + 81^{x}\right) - \frac{5}{\left(4 x - 1\right)^{2}} = 0$$
Solve:
$$x_{1} = -495921.623174481$$
$$x_{2} = -506032.64140321$$
$$x_{3} = 0.377080543676126$$
$$x_{1} = -495921.623174481$$
$$x_{2} = -506032.64140321$$
$$x_{3} = 0.377080543676126$$
This roots
$$x_{2} = -506032.64140321$$
$$x_{1} = -495921.623174481$$
$$x_{3} = 0.377080543676126$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-506032.64140321 + - \frac{1}{10}$$
=
$$-506032.74140321$$
substitute to the expression
$$\left(2 \cdot 25^{x \log{\left(5 \right)}} + 81^{x}\right) - \frac{5}{\left(4 x - 1\right)^{2}} \geq 0$$
$$- \frac{5}{\left(\left(-506032.74140321\right) 4 - 1\right)^{2}} + \left(81^{-506032.74140321} + \frac{2}{25^{506032.74140321 \log{\left(5 \right)}}}\right) \geq 0$$
                            -506032.74140321*log(5)     
-1.22037234503762e-12 + 2*25                        >= 0

but
                            -506032.74140321*log(5)    
-1.22037234503762e-12 + 2*25                        < 0

Then
$$x \leq -506032.64140321$$
no execute
one of the solutions of our inequality is:
$$x \geq -506032.64140321 \wedge x \leq -495921.623174481$$
         _____           _____  
        /     \         /
-------•-------•-------•-------
       x2      x1      x3

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -506032.64140321 \wedge x \leq -495921.623174481$$
$$x \geq 0.377080543676126$$
Solving inequality on a graph
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