Question

(x+7)(x-6)(x-11)>0 решите неравенство

Answer 1

Метод интервалов, фото

Answer 2

To solve the inequality $(x+7)(x-6)(x-11) > 0$, we need to find the intervals where the expression is positive.

To do this, we'll analyze the sign of each factor $(x+7)$, $(x-6)$, and $(x-11)$ separately. Then, we'll determine the intervals where the product of these factors is greater than zero.

Step 1: Find critical points where the factors change sign:

1. $x + 7 = 0$ => $x = -7$
2. $x - 6 = 0$ => $x = 6$
3. $x - 11 = 0$ => $x = 11$

Step 2: Create intervals based on the critical points and test each interval: We'll test each interval by picking a test point within that interval and checking the sign of $(x+7)(x-6)(x-11)$ at that point.

Interval 1: $(-\infty, -7)$ Choose $x = -8$ (a number less than -7): $((-8) + 7)((-8) - 6)((-8) - 11) = (-1)(-14)(-19) = -266$ The product is negative in this interval.

Interval 2: $(-7, 6)$ Choose $x = 0$ (a number between -7 and 6): $(0 + 7)(0 - 6)(0 - 11) = (7)(-6)(-11) = 462$ The product is positive in this interval.

Interval 3: $(6, 11)$ Choose $x = 7$ (a number between 6 and 11): $(7 + 7)(7 - 6)(7 - 11) = (14)(1)(-4) = -56$ The product is negative in this interval.

Interval 4: $(11, \infty)$ Choose $x = 12$ (a number greater than 11): $(12 + 7)(12 - 6)(12 - 11) = (19)(6)(1) = 114$ The product is positive in this interval.

Step 3: Combine the results: From the tests, we see that the expression $(x+7)(x-6)(x-11)$ is positive in two intervals: $(-7, 6)$ and $(11, \infty)$.

Step 4: Write the solution: The solution to the inequality $(x+7)(x-6)(x-11) > 0$ is: $x \in (-7, 6) \cup (11, \infty)$

So, the values of $x$ that satisfy the inequality are all real numbers in the open intervals $(-7, 6)$ and $(11, \infty)$.