Mister Exam
log3(9x)*log4(64x)/(5x^2-|x|)<=0 inequation
The solution
You have entered
[src]
log(9*x)*log(64*x)
-------------------------- <= 0
/ 2 \
log(3)*log(4)*\5*x - |x|/
$$\frac{\log{\left(9 x \right)} \log{\left(64 x \right)}}{\left(5 x^{2} - \left|{x}\right|\right) \log{\left(3 \right)} \log{\left(4 \right)}} \leq 0$$
log(9*x)*log(64*x)/((5*x^2 - |x|)*log(3)*log(4)) <= 0
Detail solution
Given the inequality:
$$\frac{\log{\left(9 x \right)} \log{\left(64 x \right)}}{\left(5 x^{2} - \left|{x}\right|\right) \log{\left(3 \right)} \log{\left(4 \right)}} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(9 x \right)} \log{\left(64 x \right)}}{\left(5 x^{2} - \left|{x}\right|\right) \log{\left(3 \right)} \log{\left(4 \right)}} = 0$$
Solve:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$x_0 = 0$$
$$\frac{\log{\left(9 \cdot 0 \right)} \log{\left(64 \cdot 0 \right)}}{\left(5 \cdot 0^{2} - \left|{0}\right|\right) \log{\left(3 \right)} \log{\left(4 \right)}} \leq 0$$
so the inequality is always executed
$$\frac{\log{\left(9 x \right)} \log{\left(64 x \right)}}{\left(5 x^{2} - \left|{x}\right|\right) \log{\left(3 \right)} \log{\left(4 \right)}} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(9 x \right)} \log{\left(64 x \right)}}{\left(5 x^{2} - \left|{x}\right|\right) \log{\left(3 \right)} \log{\left(4 \right)}} = 0$$
Solve:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$x_0 = 0$$
$$\frac{\log{\left(9 \cdot 0 \right)} \log{\left(64 \cdot 0 \right)}}{\left(5 \cdot 0^{2} - \left|{0}\right|\right) \log{\left(3 \right)} \log{\left(4 \right)}} \leq 0$$
zoo <= 0
so the inequality is always executed
Solving inequality on a graph
Rapid solution
[src]
Or(And(1/9 <= x, x < 1/5), And(x <= 1/64, 0 < x))
$$\left(\frac{1}{9} \leq x \wedge x < \frac{1}{5}\right) \vee \left(x \leq \frac{1}{64} \wedge 0 < x\right)$$
((1/9 <= x)∧(x < 1/5))∨((x <= 1/64)∧(0 < x))
Rapid solution 2
[src]
(0, 1/64] U [1/9, 1/5)
$$x\ in\ \left(0, \frac{1}{64}\right] \cup \left[\frac{1}{9}, \frac{1}{5}\right)$$
x in Union(Interval.Lopen(0, 1/64), Interval.Ropen(1/9, 1/5))