Mister Exam

# 3log11(x^2+8x-9)<4+log11(x-1)^3-log11(x+9) inequation

A inequation with variable

### The solution

You have entered [src]
     / 2          \                   3
3*log\x  + 8*x - 9/       /log(x - 1)\    log(x + 9)
------------------- < 4 + |----------|  - ----------
log(11)             \ log(11)  /     log(11)  
$$\frac{3 \log{\left(x^{2} + 8 x - 9 \right)}}{\log{\left(11 \right)}} < \left(\frac{\log{\left(x - 1 \right)}}{\log{\left(11 \right)}}\right)^{3} - \frac{\log{\left(x + 9 \right)}}{\log{\left(11 \right)}} + 4$$
3*log(x^2 + 8*x - 1*9)/log(11) < (log(x - 1*1)/log(11))^3 - log(x + 9)/log(11) + 4
Detail solution
Given the inequality:
$$\frac{3 \log{\left(x^{2} + 8 x - 9 \right)}}{\log{\left(11 \right)}} < \left(\frac{\log{\left(x - 1 \right)}}{\log{\left(11 \right)}}\right)^{3} - \frac{\log{\left(x + 9 \right)}}{\log{\left(11 \right)}} + 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 \log{\left(x^{2} + 8 x - 9 \right)}}{\log{\left(11 \right)}} = \left(\frac{\log{\left(x - 1 \right)}}{\log{\left(11 \right)}}\right)^{3} - \frac{\log{\left(x + 9 \right)}}{\log{\left(11 \right)}} + 4$$
Solve:
$$x_{1} = 249.352332301507$$
$$x_{2} = \mathtt{\text{(249.35233230150703+1.2519575680618522e-18j)}}$$
$$x_{3} = \mathtt{\text{(249.35233230150703-2.5108538302468223e-19j)}}$$
$$x_{4} = 2$$
$$x_{5} = \mathtt{\text{(249.35233230150703+1.5537333626017934e-18j)}}$$
Exclude the complex solutions:
$$x_{1} = 249.352332301507$$
$$x_{2} = 2$$
This roots
$$x_{2} = 2$$
$$x_{1} = 249.352332301507$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$1.9$$
substitute to the expression
$$\frac{3 \log{\left(x^{2} + 8 x - 9 \right)}}{\log{\left(11 \right)}} < \left(\frac{\log{\left(x - 1 \right)}}{\log{\left(11 \right)}}\right)^{3} - \frac{\log{\left(x + 9 \right)}}{\log{\left(11 \right)}} + 4$$
$$\frac{3 \log{\left(\left(-1\right) 9 + 1.9^{2} + 8 \cdot 1.9 \right)}}{\log{\left(11 \right)}} < - \frac{\log{\left(1.9 + 9 \right)}}{\log{\left(11 \right)}} + \left(\frac{\log{\left(\left(-1\right) 1 + 1.9 \right)}}{\log{\left(11 \right)}}\right)^{3} + 4$$
                       0.00116959004327434   2.3887627892351
6.85020682073181   4 - ------------------- - ---------------
---------------- <              3                log(11)
log(11)                  log (11)


one of the solutions of our inequality is:
$$x < 2$$
 _____           _____
\         /
-------ο-------ο-------
x_2      x_1

Other solutions will get with the changeover to the next point
etc.
$$x < 2$$
$$x > 249.352332301507$$