Mister Exam
(log7(9-x^2))^2-10log7(9-x^2)+21>=0 inequation
The solution
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2 / / 2\\ / 2\ |log\9 - x /| log\9 - x / |-----------| - 10*----------- + 21 >= 0 \ log(7) / log(7)
$$\left(\left(\frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right)^{2} - 10 \frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right) + 21 \geq 0$$
(log(9 - x^2)/log(7))^2 - 10*log(9 - x^2)/log(7) + 21 >= 0
Detail solution
Given the inequality:
$$\left(\left(\frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right)^{2} - 10 \frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right) + 21 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(\frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right)^{2} - 10 \frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right) + 21 = 0$$
Solve:
$$x_{1} = - \sqrt{334} i$$
$$x_{2} = \sqrt{334} i$$
$$x_{3} = - \sqrt{823534} i$$
$$x_{4} = \sqrt{823534} i$$
Exclude the complex solutions:
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
$$\left(- 10 \frac{\log{\left(9 - 0^{2} \right)}}{\log{\left(7 \right)}} + \left(\frac{\log{\left(9 - 0^{2} \right)}}{\log{\left(7 \right)}}\right)^{2}\right) + 21 \geq 0$$
so the inequality is always executed
$$\left(\left(\frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right)^{2} - 10 \frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right) + 21 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(\frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right)^{2} - 10 \frac{\log{\left(9 - x^{2} \right)}}{\log{\left(7 \right)}}\right) + 21 = 0$$
Solve:
$$x_{1} = - \sqrt{334} i$$
$$x_{2} = \sqrt{334} i$$
$$x_{3} = - \sqrt{823534} i$$
$$x_{4} = \sqrt{823534} i$$
Exclude the complex solutions:
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
x0 = 0
$$\left(- 10 \frac{\log{\left(9 - 0^{2} \right)}}{\log{\left(7 \right)}} + \left(\frac{\log{\left(9 - 0^{2} \right)}}{\log{\left(7 \right)}}\right)^{2}\right) + 21 \geq 0$$
2
log (9) 10*log(9)
21 + ------- - --------- >= 0
2 log(7)
log (7) so the inequality is always executed
Solving inequality on a graph