Question

1)-5х²<6-11х2)6х²>5х-13)-2х²+х≤-64)5х²≥4-8х​

Answer 1

Ответ:

1

$- 5 {x}^{2} < 6 - 11x \\ - 5 {x}^{2} + 11x - 6 < 0 \\ 5 {x}^{2} - 11x + 6 > 0 \\ D = 121 - 120 = 1 \\ x_1 = \frac{11 + 1}{10} = 1.2 \\ x_2 = 1 \\ + \: \: \: \: \: - \: \: \: \: \: \: \: \: \: \: + \\ - -1 - - 1.2 - - > \\ x\in( - \infty; 1)U(1.2; + \infty )$

2.

$6 {x}^{2} > 5x - 1 \\ 6 {x}^{2} - 5x + 1 > 0 \\ D= 25 - 24 = 1 \\ x_1 = \frac{5 + 1}{12} = \frac{1}{2} \\ x_2 = \frac{1}{3} \\ + \: \: \: \: - \: \: \: \: \: \: \: \: + \\ - - \frac{1}{3} - - \frac{1}{2} - - > \\ x\in( - \infty ; \frac{1}{3} )U( \frac{1}{2} ; + \infty )$

3.

$- 2 {x}^{2} + x \leqslant - 6 \\ - 2 {x}^{2} + x + 6 \leqslant 0 \\ 2 {x}^{2} - x - 6 \geqslant 0 \\ D = 1 + 48 = 49 \\ x_1 = \frac{1 + 7}{4} = 2 \\ x_2 = - 1.5 \\ + \: \: \: \: \: \: \: \: \: \: \: - \: \: \: \: \: \: \: + \\ - - ( - 1.5) - - 2 - - > \\ x\in( - \infty ; - 1.5]U[2 ;+ \infty )$

4.

$5 {x}^{2} \geqslant 4 - 8x \\ 5 {x}^{2} + 8x - 4 \geqslant 0 \\ D= 64 + 80 = 144 \\ x_1 = \frac{ - 8 + 12}{10} = 0.4 \\ x_2 = - 2 \\ + \: \: \: \: \: \: \: \: \: - \: \: \: \: \: \: \: \: \: + \\ - -( - 2) - -0.4 - - > \\ x\in( - \infty ; - 2]U[0.4; + \infty )$

Answer 2

Let's solve each of these inequalities step by step:

1. -5x² < 6 - 11x

First, move all terms to one side of the inequality to set it to zero:

-5x² + 11x - 6 < 0

Now, let's factor this quadratic equation:

-5x² + 11x - 6 = (-5x + 3)(x - 2)

Now, we have:

(-5x + 3)(x - 2) < 0

To find the solution, you can use the sign of each factor:

-5x + 3 < 0 (First factor) x - 2 > 0 (Second factor)

Solving each factor:

1. For the first factor, -5x + 3 < 0: -5x + 3 < 0 -5x < -3 x > 3/5

2. For the second factor, x - 2 > 0: x - 2 > 0 x > 2

Now, we need to combine these two conditions:

The solution is 2 < x < 3/5.

2. 6x² > 5x - 13

First, move all terms to one side of the inequality:

6x² - 5x + 13 > 0

This is a quadratic inequality. To solve it, you can use the quadratic formula. The discriminant (the value inside the square root) must be greater than zero for real solutions:

Discriminant = b² - 4ac

In this case, a = 6, b = -5, and c = 13:

Discriminant = (-5)² - 4(6)(13) = 25 - 312 = -287

Since the discriminant is negative, there are no real solutions for this inequality.

3. -2x² + x ≤ -6

First, move all terms to one side of the inequality:

-2x² + x + 6 ≤ 0

Now, we have a quadratic inequality. To solve it, you can use the quadratic formula. The discriminant must be greater than or equal to zero for real solutions:

Discriminant = b² - 4ac

In this case, a = -2, b = 1, and c = 6:

Discriminant = (1)² - 4(-2)(6) = 1 + 48 = 49

Since the discriminant is positive, there are real solutions. To find the solutions, you can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

x = (-1 ± √(49)) / (2(-2))

x = (-1 ± 7) / (-4)

Now, solve for x:

1. x = (6) / (-4) = -3/2
2. x = (-8) / (-4) = 2

So, the solutions are -3/2 and 2.

4. 5x² ≥ 4 - 8x

First, move all terms to one side of the inequality:

5x² + 8x - 4 ≥ 0

Now, we have a quadratic inequality. To solve it, you can use the quadratic formula. The discriminant must be greater than or equal to zero for real solutions:

Discriminant = b² - 4ac

In this case, a = 5, b = 8, and c = -4:

Discriminant = (8)² - 4(5)(-4) = 64 + 80 = 144

Since the discriminant is positive, there are real solutions. To find the solutions, you can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

x = (-8 ± √(144)) / (2(5))

x = (-8 ± 12) / 10

Now, solve for x:

1. x = (4) / 10 = 2/5
2. x = (-20) / 10 = -2

So, the solutions are -2 and 2/5.