(x-2)(x+10)≥0,(x+9)(x-2)(3x-9)<0,(x+6)(x-3)(x-7)≤0(x+10)(x-1)(x-5)^2≤0

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вопрос

To solve these inequalities, we need to find the critical points where the expressions equal to zero and then test the intervals between these critical points to determine when the expressions are positive, negative, or zero.

1) (x-2)(x+10)≥0

Critical points: x = 2, x = -10

Test intervals:
-∞ to -10: (-)(-) < 0
-10 to 2: (-)(+) ≥ 0
2 to ∞: (+)(+) ≥ 0

Therefore, the solution is x ≤ -10 and x ≥ 2.

2) (x+9)(x-2)(3x-9)<0

Critical points: x = -9, x = 2, x = 3

Test intervals:
-∞ to -9: (-)(-)(-) < 0
-9 to 2: (+)(-)(-) > 0
2 to 3: (+)(+)(-) < 0
3 to ∞: (+)(+)(+) > 0

Therefore, the solution is -9 < x < 2 or 3 < x.

3) (x+6)(x-3)(x-7)≤0

Critical points: x = -6, x = 3, x = 7

Test intervals:
-∞ to -6: (+)(-)(-) > 0
-6 to 3: (-)(-)(-) ≤ 0
3 to 7: (+)(+)(-) ≤ 0
7 to ∞: (+)(+)(+) > 0

Therefore, the solution is -6 ≤ x ≤ 3 or 7 ≤ x.

4) (x+10)(x-1)(x-5)^2≤0

Critical points: x = -10, x = 1, x = 5

Test intervals:
-∞ to -10: (-)(-)(-) > 0
-10 to 1: (+)(-)(-) ≤ 0
1 to 5: (+)(+)(-) ≤ 0
5 to ∞: (+)(+)(+) > 0

Therefore, the solution is -10 ≤ x ≤ 1 or x = 5.