Question

1) x^2-36/x^2+6x<0 2) x^2+2x-8x/16-x^2 (больше или равно нулю) 3) x^2-6x-7/x^2-49( меньше или равно нулю) 4) x^2-5x+4/5-x^2>0

Answer 1

1)-3x+7<=0
-3x<=-7
x>=7/3
2)4+1/2x>=x
1/2x-x>=-4
-1/2x>=-4
x<=8
3)14-8x>0
-8x>-14
x<14/8
4)2+x/3<3
x/3<3-2
x/3<1
x<3
5)-3(x+7)<(x-2)(x+1)
-3x-21<x^2+x-2x-2>0
1. Введём функцию y=x^2+2x+19, график парабола ветви вверх.
2. Нули функции:
x^2+2x+19=0
D=2^2-4*1*19=4-76=-72
D<0, нулей нет
3.Ответ x принадлежит R
6)(4x+7)-2(3x+1/2)<=0
4x+7-6x-1<=0
-2x+6<=0
-2x<=-6
x>=3

Answer 2

Let's solve each inequality step by step:

1. x^2 - 36 / (x^2 + 6x) < 0

To solve this inequality, we need to find the critical points where the expression equals zero or becomes undefined. These points are when x^2 + 6x = 0 and when x^2 + 6x = -36.

x^2 + 6x = 0 x(x + 6) = 0 x = 0 or x = -6

x^2 + 6x = -36 x^2 + 6x + 36 = 0 (x + 3)^2 = 0 x = -3

Now, we can create a sign chart to determine the intervals where the expression is positive or negative:

-∞ -6 -3 0 +∞

• | - | 0 | + | +

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From the sign chart, we can see that the expression is less than zero (-) when x is between -6 and -3, inclusive. So the solution to the inequality is -6 ≤ x ≤ -3.

2. x^2 + 2x - 8x / (16 - x^2) ≥ 0

First, we need to find the critical points where the expression equals zero or becomes undefined. These points are when x^2 + 2x - 8x = 0 and when 16 - x^2 = 0.

x^2 + 2x - 8x = 0 x^2 - 6x = 0 x(x - 6) = 0 x = 0 or x = 6

16 - x^2 = 0 x^2 = 16 x = ±4

Now, let's create a sign chart:

-∞ -4 0 4 6 +∞

• | - | 0 | + | + |

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From the sign chart, we can see that the expression is greater than or equal to zero (+) when x is less than -4 or between 0 and 4, inclusive. So the solution to the inequality is x < -4 or 0 ≤ x ≤ 4.

3. x^2 - 6x - 7 / (x^2 - 49) ≤ 0

We need to find the critical points where the expression equals zero or becomes undefined. These points are when x^2 - 6x - 7 = 0 and when x^2 - 49 = 0.

x^2 - 6x - 7 = 0 (x - 7)(x + 1) = 0 x = 7 or x = -1

x^2 - 49 = 0 (x - 7)(x + 7) = 0 x = 7 or x = -7

Let's create a sign chart:

-∞ -7 -1 7 +∞

• | - | 0 | - | - |

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From the sign chart, we can see that the expression is less than or equal to zero (-) when x is between -7 and -1, inclusive. So the solution to the inequality is -7 ≤ x ≤ -1.

4. x^2 - 5x + 4 / (5