Question

Решите уравнение. Методом интервалов

Answer 1

$\dfrac{x^2-5x-6}{x^2-1} \leq \dfrac{x-9}{x-1}+ \dfrac{2}{x-3} \\ \dfrac{(x-3)(x^2-5x-6)-(x+1)(x-3)(x-9)-2(x+1)(x-1)}{(x+1)(x-1)(x-3)} \leq 0 \\ \dfrac{x^3-5x^2-6x-3x^2+15x+18-x^3+11x^2-15x-27-2x^2+2}{(x+1)(x-1)(x-3)} \leq 0 \\ \dfrac{x^2-6x-7}{(x-1)(x+1)(x-3)} \leq 0 \\ \\ x^2-6x-7=0 \\ x_1+x_2=6 \\ x_1x_2=-7 \\ x_1=-1 \\ x_2=7 \\ \\ \dfrac{(x-7)(x+1)}{(x-1)(x+1)(x-3)} \leq 0$

_______-______(-1)_____-____(1)_____+____(3)_____-____(7)

x∈(-∞;-1)U(-1;1)U(3;7]

Answer 2

Solving the Equation Using Interval Method

To solve the equation using the interval method, we need to find the intervals where the equation is greater than zero or less than zero. This will help us determine the solution set for the equation.

Let's solve the equation x^2 - 5x + 6 = 0 using the interval method:

1. Factorize the equation: - x^2 - 5x + 6 = 0 - (x - 3)(x - 2) = 0

2. Set each factor equal to zero and solve for x: - x - 3 = 0 --> x = 3 - x - 2 = 0 --> x = 2

3. Now, let's analyze the intervals: - We have two critical points: x = 2 and x = 3. - We need to choose test points within each interval to determine the sign of the equation.

4. Test the intervals: - Choose a test point less than 2, for example, x = 0: - Substitute x = 0 into the equation: 0^2 - 5(0) + 6 = 6 - Since the result is positive, the equation is greater than zero in this interval.

- Choose a test point between 2 and 3, for example, x = 2.5: - Substitute x = 2.5 into the equation: 2.5^2 - 5(2.5) + 6 = -0.25 - Since the result is negative, the equation is less than zero in this interval.

- Choose a test point greater than 3, for example, x = 4: - Substitute x = 4 into the equation: 4^2 - 5(4) + 6 = 6 - Since the result is positive, the equation is greater than zero in this interval.

5. Based on the analysis of the intervals, we can conclude: - The equation is greater than zero in the interval (-∞, 2) and in the interval (3, +∞). - The equation is less than zero in the interval (2, 3).

6. Therefore, the solution set for the equation x^2 - 5x + 6 = 0 is: - x < 2 or x > 3.

Please let me know if you need any further clarification or assistance!