Question

1. Укажите соответствующий вывод для каждого неравенства. Обоснуйте свой ответ a) x² - 4x + 1_<0 b) 2x²-x+4>0 c) -x² + 3x - 8 _>0 d) -x² + 16 _>0 1. Неравенство не имеет решений 2. Решением неравенства является вся числовая прямая 3. Решением неравенства является одна точка 4. Решением неравенства является закрытый промежуток 5. Решением неравенства является открытый промежуток 6. Решением неравенства является объединение двух промежутков

Answer 1

Ответ:

а.

${x}^{2} - 4x + 1 \leqslant 0 \\ D = 16 - 4 = 12$

2 корня.

Ветки параболы направлены вверх, у<=0 на одном промежутке

4. Закрытый промежуток

b.

$2 {x}^{2} - x + 4 > 0 \\ D = 1 - 32 < 0$

корней нет

Парабола выше оси ОХ, все у>0

2. Вся числовая прямая

с.

$- {x}^{2} + 3x - 8 \geqslant 0 \\ {x}^{2} - 3x + 8 \leqslant 0 \\ D= 9 - 32 < 0$

корней нет

Парабола выше ОХ. Отрицательных у нет

1. Нет решения

d.

$- x {}^{2} + 16 \geqslant 0 \\ {x}^{2} - 16 \leqslant 0 \\ (x - 4)(x +4) \leqslant 0 \\ + \: \: \: \: \: \: \: - \: \: \: \: \: \: + \\ - -( - 4) - -4 - - > \\ x\in[ - 4;4]$

4. Закрытый промежуток

Answer 2

a) x² - 4x + 1 < 0

To determine the corresponding conclusion for this inequality, we can solve it by factoring or by using the quadratic formula. Let's factor the quadratic expression:

(x - 1)(x - 1) < 0

The quadratic expression is a perfect square trinomial, and when multiplied together, it will always be positive or zero. Therefore, the inequality has no solutions.

Conclusion: 1. The inequality has no solutions.

b) 2x² - x + 4 > 0

To find the conclusion for this inequality, we can solve it using the quadratic formula. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

For this inequality, the quadratic equation is:

2x² - x + 4 = 0

Using the quadratic formula, we can find the roots of the equation:

x = (1 ± √(-31)) / 4

Since the discriminant (b² - 4ac) is negative, the quadratic equation has no real solutions. Therefore, the inequality is always true for all real values of x.

Conclusion: 2. The solution to the inequality is the entire real number line.

c) -x² + 3x - 8 > 0

To determine the conclusion for this inequality, we can solve it by factoring or by using the quadratic formula. Let's factor the quadratic expression:

(-x + 4)(x - 2) > 0

To find the values of x that satisfy the inequality, we can set each factor separately to zero and determine the sign of the inequality in each interval:

x - 4 > 0 => x > 4 x - 2 > 0 => x > 2

Since the inequality is greater than zero, the solution is the union of the intervals (4, ∞) and (2, ∞).

Conclusion: 6. The solution to the inequality is the union of two open intervals.

d) -x² + 16 > 0

To find the conclusion for this inequality, we can solve it by factoring or by using the quadratic formula. Let's factor the quadratic expression:

(-x - 4)(x + 4) > 0

To find the values of x that satisfy the inequality, we can set each factor separately to zero and determine the sign of the inequality in each interval:

-x - 4 > 0 => x < -4 x + 4 > 0 => x > -4

Since the inequality is greater than zero, the solution is the union of the intervals (-∞, -4) and (-4, ∞).

Conclusion: 6. The solution to the inequality is the union of two intervals.