[tex] \frac{(x - 6)( x+ 5) ^{2}(x - 10) }{x - 20} > 0[/tex]

Предыдущий
вопрос Следующий
вопрос

To solve this inequality, we first need to find the critical points where the expression is equal to 0 or undefined.

Setting the numerator equal to 0 gives us the critical points:

(x - 6 = 0 \Rightarrow x = 6)(x + 5 = 0 \Rightarrow x = -5)(x - 10 = 0 \Rightarrow x = 10)

Setting the denominator equal to 0 gives us:
(x - 20 = 0 \Rightarrow x = 20)

So, the critical points are (x = -5), (x = 6), (x = 10) and (x = 20).

Now we can create a number line and test the intervals (from left to right) formed by these critical points in the original inequality.

For (x < -5): Pick (x = -6), then the expression becomes (\frac{(-6 - 6)(-6 + 5)^2(-6 - 10)}{-6 -20} = 300 > 0).For (-5 < x < 6): Pick (x = 0), then the expression becomes (\frac{(0 - 6)(0 + 5)^2(0 - 10)}{0 - 20} = 450 < 0).For (6 < x < 10): Pick (x = 7), then the expression becomes (\frac{(7 - 6)(7 + 5)^2(7 - 10)}{7 - 20} = -180 > 0).For (10 < x < 20): Pick (x = 15), then the expression becomes (\frac{(15 - 6)(15 + 5)^2(15 - 10)}{15 - 20} = 360 > 0).For (x > 20): Pick (x = 21), then the expression becomes (\frac{(21 - 6)(21 + 5)^2(21 - 10)}{21 - 20} = 684 > 0).

Therefore, the solution to the inequality is (x \in (-\infty, -5) \cup (6, 10) \cup (10, 20) \cup (20, \infty)).