Mister Exam

# 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))
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 = 4, x <= 1|
|   |    3   \/     54        18               ___________________  3           _______________            3              |               |
|   |                                         /          ________              /          ____                            |               |
|   |                                        /  2189   \/ 691485              /  47   3*\/ 93                             |               |
|   |                                   9*3 /   ---- + ----------        3*3 /   -- + --------                            |               |
\   \                                     \/     54        18              \/    2       2                                /               /
$$\left(x < - \frac{71}{9 \sqrt[3]{\frac{2189}{54} + \frac{\sqrt{691485}}{18}}} + \frac{4}{3} + \sqrt[3]{\frac{2189}{54} + \frac{\sqrt{691485}}{18}} \wedge - \frac{\sqrt[3]{\frac{3 \sqrt{93}}{2} + \frac{47}{2}}}{3} - \frac{7}{3 \sqrt[3]{\frac{3 \sqrt{93}}{2} + \frac{47}{2}}} + \frac{17}{3} < x\right) \vee x = 4 \vee x \leq 1$$
(x = 4))∨(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 (-- - -------------------- - -----------------------, - - -------------------------- + -----------------------------) U {4}
3         _______________              6             3        _____________________                 6
3 /          ____                                   3 /            ________
3*\/  47 + 3*\/ 93                                  3*\/  2189 + 3*\/ 691485                                         
$$x\ in\ \left(-\infty, 1\right] \cup \left(- \frac{2^{\frac{2}{3}} \sqrt[3]{3 \sqrt{93} + 47}}{6} - \frac{7 \sqrt[3]{2}}{3 \sqrt[3]{3 \sqrt{93} + 47}} + \frac{17}{3}, - \frac{71 \sqrt[3]{2}}{3 \sqrt[3]{2189 + 3 \sqrt{691485}}} + \frac{4}{3} + \frac{2^{\frac{2}{3}} \sqrt[3]{2189 + 3 \sqrt{691485}}}{6}\right) \cup \left\{4\right\}$$
x in Union(FiniteSet(4), 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))
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