# log(1/3,(4-x)(x^2+29))<=log(1/3,(x^2-10x+24)(7-x)) inequation

A inequation with variable

### The solution

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   /             / 2     \\       /     / 2            \        \
log\1/3, (4 - x)*\x  + 29// <= log\1/3, \x  - 10*x + 24/*(7 - x)/
$$\log{\left(\frac{1}{3} \right)} \leq \log{\left(\frac{1}{3} \right)}$$
log(1/3, (4 - x)*(x^2 + 29)) <= log(1/3, (7 - x)*(x^2 - 10*x + 24))
Detail solution
Given the inequality:
$$\log{\left(\frac{1}{3} \right)} \leq \log{\left(\frac{1}{3} \right)}$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{1}{3} \right)} = \log{\left(\frac{1}{3} \right)}$$
Solve:
$$x_{1} = 1$$
$$x_{1} = 1$$
This roots
$$x_{1} = 1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1$$
=
$$\frac{9}{10}$$
substitute to the expression
$$\log{\left(\frac{1}{3} \right)} \leq \log{\left(\frac{1}{3} \right)}$$
$$\log{\left(\frac{1}{3} \right)} \leq \log{\left(\frac{1}{3} \right)}$$
 -log(3)       -log(3)
----------    ----------
/92411\ <=    /96441\
log|-----|    log|-----|
\ 1000/       \ 1000/

the solution of our inequality is:
$$x \leq 1$$
 _____
\
-------•-------
x_1
Solving inequality on a graph
Rapid solution [src]
  /   /                                                                                                  _______________    \        \
|   |                                                                                                 /          ____     |        |
|   |             ___________________                                                                /  47   3*\/ 93      |        |
|   |            /          ________                                                              3 /   -- + --------     |        |
|   |    4      /  2189   \/ 691485                 71              17             7              \/    2       2         |        |
Or|And|x < - + 3 /   ---- + ----------  - --------------------------, -- - ---------------------- - -------------------- < x|, x <= 1|
|   |    3   \/     54        18               ___________________  3           _______________            3              |        |
|   |                                         /          ________              /          ____                            |        |
|   |                                        /  2189   \/ 691485              /  47   3*\/ 93                             |        |
|   |                                   9*3 /   ---- + ----------        3*3 /   -- + --------                            |        |
\   \                                     \/     54        18              \/    2       2                                /        /
$$\left(x < - \frac{71}{9 \sqrt{\frac{2189}{54} + \frac{\sqrt{691485}}{18}}} + \frac{4}{3} + \sqrt{\frac{2189}{54} + \frac{\sqrt{691485}}{18}} \wedge - \frac{\sqrt{\frac{3 \sqrt{93}}{2} + \frac{47}{2}}}{3} - \frac{7}{3 \sqrt{\frac{3 \sqrt{93}}{2} + \frac{47}{2}}} + \frac{17}{3} < x\right) \vee x \leq 1$$
(x <= 1)∨((x < 4/3 + (2189/54 + sqrt(691485)/18)^(1/3) - 71/(9*(2189/54 + sqrt(691485)/18)^(1/3)))∧(17/3 - 7/(3*(47/2 + 3*sqrt(93)/2)^(1/3)) - (47/2 + 3*sqrt(93)/2)^(1/3)/3 < x))
Rapid solution 2 [src]
                                                _______________                                           _____________________
3 ___           2/3 3 /          ____                   3 ___             2/3 3 /            ________
17         7*\/ 2           2   *\/  47 + 3*\/ 93    4            71*\/ 2             2   *\/  2189 + 3*\/ 691485
(-oo, 1] U (-- - -------------------- - -----------------------, - - -------------------------- + -----------------------------)
3         _______________              6             3        _____________________                 6
3 /          ____                                   3 /            ________
3*\/  47 + 3*\/ 93                                  3*\/  2189 + 3*\/ 691485                                   
$$x\ in\ \left(-\infty, 1\right] \cup \left(- \frac{2^{\frac{2}{3}} \sqrt{3 \sqrt{93} + 47}}{6} - \frac{7 \cdot \sqrt{2}}{3 \sqrt{3 \sqrt{93} + 47}} + \frac{17}{3}, - \frac{71 \cdot \sqrt{2}}{3 \sqrt{2189 + 3 \sqrt{691485}}} + \frac{4}{3} + \frac{2^{\frac{2}{3}} \sqrt{2189 + 3 \sqrt{691485}}}{6}\right)$$
x in Union(Interval(-oo, 1), Interval.open(-2^(2/3)*(3*sqrt(93) + 47)^(1/3)/6 - 7*2^(1/3)/(3*(3*sqrt(93) + 47)^(1/3)) + 17/3, -71*2^(1/3)/(3*(2189 + 3*sqrt(691485))^(1/3)) + 4/3 + 2^(2/3)*(2189 + 3*sqrt(691485))^(1/3)/6))