Question

2) (x-7)(x + 8)(x - 12) >0;​

Answer 1

Решаем методом интервалов.

Пусть f(x)=(x-7)(x + 8)(x - 12)

Найдем нули функции.

f(x)=0      ⇒   (x-7)(x + 8)(x - 12)=0

x=7;  x=-8; x=12

Проходя через нуль функции график переходит из одной полуплоскости в другую

Поэтому находим знаки функции на интервалах:

_____ (-8) _____ (7) _____ (12) ____

f(13)=(13-7)(13 + 8)(13 - 12) >0  ставим справа от 12 +

_____ (-8) _____ (7) _____ (12) __+__

Далее знаки чередуются:

__-___ (-8) __+___ (7) __-___ (12) __+__

О т в е т. (-8;7) U(12;+∞)

Answer 2

To determine the solution to the inequality (x - 7)(x + 8)(x - 12) > 0, we need to find the intervals on the number line where the expression is greater than zero.

To solve this, we'll use the concept of interval notation and the sign of the expression within each interval.

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

x - 7 = 0 => x = 7 x + 8 = 0 => x = -8 x - 12 = 0 => x = 12

Now, let's consider the intervals between these critical points and evaluate the sign of the expression within each interval:

For x < -8: Plugging in a test value, let's say x = -9: (-9 - 7)(-9 + 8)(-9 - 12) = (-16)(-1)(-21) = 336 Since the expression is positive, the interval x < -8 satisfies the inequality.

For -8 < x < 7: Plugging in a test value, let's say x = 0: (0 - 7)(0 + 8)(0 - 12) = (-7)(8)(-12) = 672 Since the expression is positive, the interval -8 < x < 7 satisfies the inequality.

For 7 < x < 12: Plugging in a test value, let's say x = 10: (10 - 7)(10 + 8)(10 - 12) = (3)(18)(-2) = -108 Since the expression is negative, the interval 7 < x < 12 does not satisfy the inequality.

For x > 12: Plugging in a test value, let's say x = 13: (13 - 7)(13 + 8)(13 - 12) = (6)(21)(1) = 126 Since the expression is positive, the interval x > 12 satisfies the inequality.

Therefore, the solution to the inequality (x - 7)(x + 8)(x - 12) > 0 is: x < -8 or -8 < x < 7 or x > 12.