Question

(x-3)(2x + 5) < (2x - 6) (2x - 1)​

Answer 1

Ответ:

Пошаговое объяснение:

(x-3)(2x+5)<(2x-6)(2x-1)

2x²+5x-6x-15<4x²-2x-12x+6

2x²-x-15<4x²-14x+6

4x²-14x+6-2x²+x+15>0

2x²-13x+21>0

Допустим:

2x²-13x+21=0; D=169-168=1

x₁=(13-1)4=12/4=3

x₂=(13+1)/4=14/4=7/2=3,5

Нанесём точки на ось x. Выбираем точку из любого интервала для проверки знака на интервале, например 0:

2·0²-13·0+21>0; 21>0⇒знак на интервале, в котором лежит точка 0 будет +.

 

 +                  -                    +

----------.------------------.---------------------> x

          3                   3,5

(-∞<x₁<3)∨(3,5<x₂<+∞)

Ответ: x∈(-∞; 3)∪(3,5; +∞).

Answer 2

To solve the inequality (x-3)(2x + 5) < (2x - 6)(2x - 1), we can expand both sides and simplify:

Expanding the left side: (x-3)(2x + 5) = 2x(x) + 5(x) - 3(2x) - 3(5) = 2x^2 + 5x - 6x - 15 = 2x^2 - x - 15

Expanding the right side: (2x - 6)(2x - 1) = 4x^2 - 2x - 12x + 6 = 4x^2 - 14x + 6

Now, we have the inequality: 2x^2 - x - 15 < 4x^2 - 14x + 6

Rearranging the terms, we get: 0 < 2x^2 - 14x + 21

To solve this inequality, we can find the roots of the quadratic equation 2x^2 - 14x + 21 = 0.

Factoring the quadratic equation, we have: 2x^2 - 14x + 21 = (2x - 3)(x - 7)

Setting each factor equal to zero and solving for x: 2x - 3 = 0 => 2x = 3 => x = 3/2 x - 7 = 0 => x = 7

These are the critical points of the inequality. We can test the intervals between the critical points and beyond to determine the solution.

1. For x < 3/2: Pick a test point, e.g., x = 0: 2(0)^2 - 14(0) + 21 = 21 > 0

2. For 3/2 < x < 7: Pick a test point, e.g., x = 2: 2(2)^2 - 14(2) + 21 = 1 > 0

3. For x > 7: Pick a test point, e.g., x = 8: 2(8)^2 - 14(8) + 21 = -67 < 0

Based on the tests, the solution to the inequality (x-3)(2x + 5) < (2x - 6)(2x - 1) is: x < 3/2 or 3/2 < x < 7.