Question

(x - 6) (x + 1) < 0;​

Answer 1

Ответ:

Объяснение:

(x - 6) (x + 1) < 0;​

(x - 6)< 0;​  =>x < 6

(x + 1) > 0;  =>x  >- 1;   => x∈(-1;6)

или

(x - 6)> 0;​  =>x > 6

(x + 1) < 0;​ =>  x  <- 1;    x∈(-∞;-1)∩(6;∞)= ∅

отв x∈(-1;6)

Answer 2

To solve the inequality $(x - 6)(x + 1) < 0$, you can use the concept of interval notation and the properties of the product of two factors being less than zero.

1. First, find the critical points by setting each factor equal to zero and solving for $x$: $x - 6 = 0$ gives $x = 6$, and $x + 1 = 0$ gives $x = -1$.

2. These critical points divide the number line into three intervals: $(-∞, -1)$, $(-1, 6)$, and $(6, ∞)$.

3. Choose a test point from each interval and plug it into the original inequality $(x - 6)(x + 1) < 0$ to determine the sign of the expression in each interval.

   • For the interval $(-∞, -1)$, choose $x = -2$: $((-2) - 6)((-2) + 1) = (-8)(-1) = 8$, which is positive.

   • For the interval $(-1, 6)$, choose $x = 0$: $(0 - 6)(0 + 1) = (-6)(1) = -6$, which is negative.

   • For the interval $(6, ∞)$, choose $x = 7$: $(7 - 6)(7 + 1) = (1)(8) = 8$, which is positive.

4. Now, consider the signs of $(x - 6)(x + 1)$ in each interval:

   • In $(-∞, -1)$, $(x - 6)(x + 1)$ is positive.
   • In $(-1, 6)$, $(x - 6)(x + 1)$ is negative.
   • In $(6, ∞)$, $(x - 6)(x + 1)$ is positive.

5. The inequality $(x - 6)(x + 1) < 0$ is satisfied when $(x - 6)(x + 1)$ is negative, which occurs in the interval $(-1, 6)$.

So, the solution to the inequality is $x \in (-1, 6)$ in interval notation. This means that the values of $x$ that satisfy the inequality are between -1 and 6, exclusive.