Question

Розв'яжіть нерівність методом інтервалів 1) (х-1)(х+3)<0 2) (х+5)(2х-8)> або=0

Answer 1

Ответ:

решение смотри на фотографии

Объяснение:

Answer 2

1. (х-1)(х+3) < 0:

To solve this inequality using the interval method, we need to find the intervals where the expression (x-1)(x+3) is less than zero (negative).

Step 1: Find the critical points by setting the expression equal to zero: (x - 1)(x + 3) = 0

Setting each factor equal to zero, we have: x - 1 = 0 => x = 1 x + 3 = 0 => x = -3

Step 2: Plot the critical points on a number line:

diffCopy code
-3    1
 |-----|-----|

Step 3: Pick a test point from each interval:

For the interval (-∞, -3), we can choose x = -4. For the interval (-3, 1), we can choose x = 0. For the interval (1, +∞), we can choose x = 2.

Step 4: Evaluate the expression (x - 1)(x + 3) using the test points:

For x = -4: (-4 - 1)(-4 + 3) = (-5)(-1) = 5 > 0 For x = 0: (0 - 1)(0 + 3) = (-1)(3) = -3 < 0 For x = 2: (2 - 1)(2 + 3) = (1)(5) = 5 > 0

Step 5: Determine the intervals where the expression is less than zero:

The expression (x - 1)(x + 3) is negative (less than zero) in the interval (-3, 1).

Therefore, the solution to the inequality (x - 1)(x + 3) < 0 is: -3 < x < 1.

2. (х+5)(2х-8) > 0 or equal to zero:

To solve this inequality using the interval method, we need to find the intervals where the expression (x + 5)(2x - 8) is greater than zero (positive) or equal to zero.

Step 1: Find the critical points by setting the expression equal to zero: (x + 5)(2x - 8) = 0

Setting each factor equal to zero, we have: x + 5 = 0 => x = -5 2x - 8 = 0 => x = 4

Step 2: Plot the critical points on a number line:

diffCopy code
-5    4
 |-----|-----|

Step 3: Pick a test point from each interval:

For the interval (-∞, -5), we can choose x = -6. For the interval (-5, 4), we can choose x = 0. For the interval (4, +∞), we can choose x = 5.

Step 4: Evaluate the expression (x + 5)(2x - 8) using the test points:

For x = -6: (-6 + 5)(2(-6) - 8) = (-1)(-20) = 20 > 0 For x = 0: (0 + 5)(2(0) - 8) = (5)(-8) = -40 < 0 For x = 5: (5 + 5)(2(5) - 8) = (10)(2) = 20 > 0

Step 5: Determine the intervals where the expression is greater than zero:

The expression (x + 5)(2x - 8) is positive (greater than zero) in the intervals (-∞, -5) and (4, +∞). It is also equal to zero at x = 4.

Therefore, the solution to the inequality (x + 5)(2x - 8) > 0 or equal to zero is: x < -5 or x > 4.