Question

-(х-1)(5-х)(х+20)>0​

Answer 1

Ответ:

Объяснение:

Умножаем на -1:

(x-1)(5-x)(x+20)<0;

Корни:-20; 1; 5.

_____-20_____1______5_______

    +              -              +                -

x∈(-20;1)∪(5;+∞)

Answer 2

To solve the inequality $-(x-1)(5-x)(x+20) > 0$, we need to find the intervals where the expression on the left-hand side is greater than zero.

First, let's find the critical points by setting each factor in the expression to zero and solving for $x$:

1. $x - 1 = 0 \Rightarrow x = 1$
2. $5 - x = 0 \Rightarrow x = 5$
3. $x + 20 = 0 \Rightarrow x = -20$

Now, let's plot these critical points on a number line:

pythonCopy code
... -20 ........ 1 ........ 5 ...

Now, we need to test the intervals created by these critical points.

Pick a value from each interval and substitute it into the original inequality to determine the sign of the expression in that interval:

1. For $x < -20$: Choose $x = -21$ $-(-21 - 1)(5 - (-21))( -21 + 20) = -(-22)(26)(-1) = 572 > 0$ (positive)

2. For $-20 < x < 1$: Choose $x = 0$ $-((0) - 1)(5 - (0))( (0) + 20) = -( -1)(5)(20) = -100 < 0$ (negative)

3. For $1 < x < 5$: Choose $x = 2$ $-(2 - 1)(5 - 2)(2 + 20) = -(1)(3)(22) = -66 < 0$ (negative)

4. For $x > 5$: Choose $x = 6$ $-(6 - 1)(5 - 6)(6 + 20) = -(5)(-1)(26) = 130 > 0$ (positive)

Now, based on the signs in each interval, we can see that the expression $-(x-1)(5-x)(x+20)$ is positive in the intervals $(-20, 1)$ and $(5, \infty)$.

Thus, the solution to the inequality is:

$-20 < x < 1$ OR $x > 5$

In interval notation, this is written as:

$(-20, 1) \cup (5, \infty)$