Question

Решить неравенства: 1) x(x-2)(x+7) ≥ 0;2) -(x²-4)(x+5) ≤ 0;3) (9-x²)(5x+10) > 0;

Answer 1

Ответ:

вот, держи, думаю понятно

Answer 2

To solve these inequalities, we need to find the intervals of x that satisfy the given conditions. The approach is to determine the sign of each factor and then combine them to find the intervals where the inequality holds true. Let's solve each inequality step by step:

1. x(x-2)(x+7) ≥ 0

Step 1: Find the critical points where the factors are equal to zero. x = 0, x = 2, x = -7

Step 2: Plot these critical points on the number line and pick a test point from each interval.

Test points: For x < -7, pick x = -8 For -7 < x < 0, pick x = -1 For 0 < x < 2, pick x = 1 For x > 2, pick x = 3

Step 3: Evaluate the sign of each factor and the whole expression at the test points.

Signs: x: - - + + x - 2: - - - + x + 7: - + + + Expression: + - - +

Step 4: Determine the intervals where the expression is greater than or equal to zero.

The solution is: x ∈ (-∞, -7] U [0, 2].

2. -(x²-4)(x+5) ≤ 0

Step 1: Find the critical points where the factors are equal to zero. x² - 4 = 0 --> x = ±2 x + 5 = 0 --> x = -5

Step 2: Plot these critical points on the number line and pick a test point from each interval.

Test points: For x < -5, pick x = -6 For -5 < x < -2, pick x = -3 For -2 < x < 2, pick x = 0 For x > 2, pick x = 3

Step 3: Evaluate the sign of each factor and the whole expression at the test points.

Signs: x² - 4: + + - - x + 5: - - + + Expression: + - + -

Step 4: Determine the intervals where the expression is less than or equal to zero.

The solution is: x ∈ [-5, -2] U [2, ∞).

3. (9-x²)(5x+10) > 0

Step 1: Find the critical points where the factors are equal to zero. 9 - x² = 0 --> x = ±3 5x + 10 = 0 --> x = -2

Step 2: Plot these critical points on the number line and pick a test point from each interval.

Test points: For x < -3, pick x = -4 For -3 < x < -2, pick x = -2.5 For -2 < x < 3, pick x = 0 For x > 3, pick x = 4

Step 3: Evaluate the sign of each factor and the whole expression at the test points.

Signs: 9 - x²: - - + + 5x + 10: 0 - + + Expression: 0 + + +

Step 4: Determine the intervals where the expression is greater than zero (exclude zero itself).

The solution is: x ∈ (-∞, -3) U (3, ∞).

In summary:

1. x ∈ (-∞, -7] U [0, 2]
2. x ∈ [-5, -2] U [2, ∞)
3. x ∈ (-∞, -3) U (3, ∞)