Question

(2x+7)( 3x-4)(x+5) больше 0 решите неравенство методом интервалов

Answer 1

Найдем значения Х, которые обнулят выражения, стоящие в скобках:
1)2x+7=0
   2x=-7
   x=-3,5
2)3x-4=0
   3x=4
   x=4/3
3)x+5=0
   x=-5
Отметим эти точки на числовой прямой:

_____-_______(-5)____+______(-3,5)_______-____(4/3)_____+__
                             ////////////////////////                                    ///////////////////

Ответ: x e (-5;-3,5) U (4/3; + беск.)

Answer 2

Solution to the Inequality (2x+7)(3x-4)(x+5) > 0 using Interval Method

To solve the inequality (2x+7)(3x-4)(x+5) > 0 using the interval method, we need to find the intervals where the expression is greater than zero.

Let's break down the problem step by step:

1. Find the critical points: - The critical points are the values of x where the expression (2x+7)(3x-4)(x+5) equals zero. To find these points, we set each factor equal to zero and solve for x: - 2x + 7 = 0 => x = -7/2 - 3x - 4 = 0 => x = 4/3 - x + 5 = 0 => x = -5

2. Create intervals: - We create intervals using the critical points and divide the number line into four regions: - Region 1: x < -7/2 - Region 2: -7/2 < x < 4/3 - Region 3: 4/3 < x < -5 - Region 4: x > -5

3. Test the intervals: - We choose a test point from each interval and substitute it into the expression (2x+7)(3x-4)(x+5) to determine whether the expression is positive or negative in that interval. - Let's choose the test points: - Test point for Region 1: x = -8 - Test point for Region 2: x = 0 - Test point for Region 3: x = 2 - Test point for Region 4: x = 6

4. Evaluate the expression: - We substitute the test points into the expression (2x+7)(3x-4)(x+5) and determine whether the expression is positive or negative: - For x = -8: (2(-8)+7)(3(-8)-4)(-8+5) = (-9)(-28)(-3) = -756 < 0 - For x = 0: (2(0)+7)(3(0)-4)(0+5) = (7)(-4)(5) = -140 < 0 - For x = 2: (2(2)+7)(3(2)-4)(2+5) = (11)(2)(7) = 154 > 0 - For x = 6: (2(6)+7)(3(6)-4)(6+5) = (19)(14)(11) = 2938 > 0

5. Determine the intervals where the expression is greater than zero: - From the evaluations above, we can see that the expression (2x+7)(3x-4)(x+5) is greater than zero in Region 3 and Region 4.

6. Final solution: - Therefore, the solution to the inequality (2x+7)(3x-4)(x+5) > 0 using the interval method is: - x ∈ (4/3, -5) U (-5, ∞).

Please let me know if you need any further clarification.