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64/(2^(x+3))<=0,125^x-7

  • How to use it?

  • Inequation:
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  • cos((x/2)+(pi/3))
  • |x-1|-(6/|x-1|)<1
  • 64/(2^(x+3))<=0,125^x-7 64/(2^(x+3))<=0,125^x-7

  • Identical expressions

  • sixty-four /(two ^(x+ three))<= zero , one hundred and twenty-five ^x- seven
  • 64 divide by (2 to the power of (x plus 3)) less than or equal to 0,125 to the power of x minus 7
  • sixty minus four divide by (two to the power of (x plus three)) less than or equal to zero , one hundred and twenty minus five to the power of x minus seven
  • 64/(2(x+3))<=0,125x-7
  • 64/2x+3<=0,125x-7
  • 64/2^x+3<=0,125^x-7
  • 64/(2^(x+3))<=O,125^x-7
  • 64 divide by (2^(x+3))<=0,125^x-7

  • Similar expressions

  • 64/(2^(x-3))<=0,125^x-7
  • 64/(2^(x+3))<=0,125^x+7
Solving inequalities/ 64/(2^(x+3))<=0,125^x-7

64/(2^(x+3))<=0,125^x-7 inequation

A inequation with variable

The solution

You have entered [src] 
  64       -x    
------ <= 8   - 7
 x + 3           
2
$$\frac{64}{2^{x + 3}} \leq \left(-1\right) 7 + \left(\frac{1}{8}\right)^{x}$$ 
64/(2^(x + 3)) <= -1*7 + (1/8)^x
Detail solution
Given the inequality:
$$\frac{64}{2^{x + 3}} \leq \left(-1\right) 7 + \left(\frac{1}{8}\right)^{x}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{64}{2^{x + 3}} = \left(-1\right) 7 + \left(\frac{1}{8}\right)^{x}$$
Solve:
Given the equation:
$$\frac{64}{2^{x + 3}} = \left(-1\right) 7 + \left(\frac{1}{8}\right)^{x}$$
or
$$\left(7 - \left(\frac{1}{8}\right)^{x}\right) + \frac{64}{2^{x + 3}} = 0$$
Do replacement
$$v = \left(\frac{1}{8}\right)^{x}$$
we get
$$64 \cdot 2^{- x - 3} + 7 - 8^{- x} = 0$$
or
$$64 \cdot 2^{- x - 3} + 7 - 8^{- x} = 0$$
do backward replacement
$$\left(\frac{1}{8}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(8 \right)}}$$
$$x_{1} = \frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}}$$
$$x_{2} = \frac{\log{\left(\frac{1}{14} + \frac{\sqrt{29}}{14} \right)} + i \pi}{\log{\left(2 \right)}}$$
$$x_{3} = \frac{i \pi}{\log{\left(2 \right)}}$$
Exclude the complex solutions:
$$x_{1} = \frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}}$$
This roots
$$x_{1} = \frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}} - \frac{1}{10}$$
=
$$\frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}} - \frac{1}{10}$$
substitute to the expression
$$\frac{64}{2^{x + 3}} \leq \left(-1\right) 7 + \left(\frac{1}{8}\right)^{x}$$
$$\frac{64}{2^{\left(\frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}} - \frac{1}{10}\right) + 3}} \leq \left(-1\right) 7 + \left(\frac{1}{8}\right)^{\frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}} - \frac{1}{10}}$$
                         /       ____\                             /       ____\
      29   -log(14) + log\-1 + \/ 29 /          1    -log(14) + log\-1 + \/ 29 /
    - -- - --------------------------- <=       -- - ---------------------------
      10              log(2)                    10              log(2)          
64*2                                      -7 + 8

the solution of our inequality is:
$$x \leq \frac{- \log{\left(14 \right)} + \log{\left(-1 + \sqrt{29} \right)}}{\log{\left(2 \right)}}$$
 _____          
      \    
-------•-------
       x_1
Solving inequality on a graph
Rapid solution [src] 
   /        /         ____\         \
   |        |  1    \/ 29 |         |
   |     log|- -- + ------|         |
   |        \  14     14  /         |
And|x <= ------------------, -oo < x|
   \           log(2)               /
$$x \leq \frac{\log{\left(- \frac{1}{14} + \frac{\sqrt{29}}{14} \right)}}{\log{\left(2 \right)}} \wedge -\infty < x$$ 
(-oo < x)∧(x <= log(-1/14 + sqrt(29)/14)/log(2))
Rapid solution 2 [src] 
         /         ____\ 
         |  1    \/ 29 | 
      log|- -- + ------| 
         \  14     14  / 
(-oo, ------------------]
            log(2)
$$x\ in\ \left(-\infty, \frac{\log{\left(- \frac{1}{14} + \frac{\sqrt{29}}{14} \right)}}{\log{\left(2 \right)}}\right]$$ 
x in Interval(-oo, log(-1/14 + sqrt(29)/14)/log(2))
The graph
64/(2^(x+3))<=0,125^x-7 inequation
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