Question

Решите неравенство (х+4) (х-1) (х-2)≥0​

Answer 1

х+4≥0.........х-1≥0........х-2≥0

х≥0-4.........х≥0+1........х≥0+2

х≥-4..........х≥1..........х≥2

Ответ: х1≥-4, х2≥1, х3≥2

Проверка:

(-4+4)(1-1)(2-2)≥0

0≥0

Answer 2

To solve the inequality $(x + 4)(x - 1)(x - 2) \geq 0$, we need to find the intervals for which the expression is non-negative (greater than or equal to zero).

Step 1: Find critical points by setting each factor equal to zero and solving for $x$:

$(x + 4) = 0 \implies x = -4$

$(x - 1) = 0 \implies x = 1$

$(x - 2) = 0 \implies x = 2$

Step 2: Create a number line and mark the critical points:

diffCopy code
-4     1     2
|------|------|------|

Step 3: Pick a test point in each of the intervals created by the critical points and evaluate the expression $(x + 4)(x - 1)(x - 2)$ for each test point to determine the sign:

Test $x = -5$ (a number less than -4): $(-5 + 4)(-5 - 1)(-5 - 2) = (-1)(-6)(-7) = 42$ (positive)

Test $x = 0$ (a number between -4 and 1): $(0 + 4)(0 - 1)(0 - 2) = (4)(-1)(-2) = 8$ (positive)

Test $x = 1.5$ (a number between 1 and 2): $(1.5 + 4)(1.5 - 1)(1.5 - 2) = (5.5)(0.5)(-0.5) = -1.375$ (negative)

Test $x = 3$ (a number greater than 2): $(3 + 4)(3 - 1)(3 - 2) = (7)(2)(1) = 14$ (positive)

Step 4: Determine the intervals where the expression is non-negative (greater than or equal to zero):

The expression is non-negative (greater than or equal to zero) in the intervals: $(-4, 1]$ and $[2, +\infty)$

Therefore, the solution to the inequality is $x \in (-4, 1] \cup [2, +\infty)$.