Question

(4 + x)(2 - x) + (1 - x) > 0​

Answer 1

Ну вот и всё . пожалуйста

Answer 2

To solve the inequality (4 + x)(2 - x) + (1 - x) > 0, we need to find the values of x that make the expression on the left-hand side greater than zero.

Let's simplify the expression first:

(4 + x)(2 - x) + (1 - x) > 0

Expand the products:

(8 - 4x + 2x - x^2) + (1 - x) > 0

Combine like terms:

(8 - 2x - x^2) + (1 - x) > 0

Now, combine the terms in the parentheses:

9 - 2x - x^2 - x > 0

To proceed, we need to set the inequality to zero. We can do this by moving all terms to the left side of the inequality:

9 - 2x - x^2 - x - 0 > 0

Now, rearrange the terms:

-x^2 - 3x + 9 > 0

The next step is to find the critical points of the inequality. Critical points occur when the left-hand side expression is equal to zero. To find these points, set the expression equal to zero:

-x^2 - 3x + 9 = 0

Now, we need to solve for x. However, this quadratic equation doesn't factor easily, so we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = -1, b = -3, and c = 9.

x = (3 ± √(9 - 4(-1)(9))) / 2(-1)

x = (3 ± √(9 + 36)) / -2

x = (3 ± √45) / -2

x = (3 ± 6.708) / -2

Now we have two potential solutions for x:

1. x = (3 + 6.708) / -2 ≈ -4.354
2. x = (3 - 6.708) / -2 ≈ 1.354

Now, we'll analyze the inequality in different intervals defined by these critical points and evaluate whether the expression is greater than zero or not:

1. When x < -4.354: Substitute x = -5 (for example): -(-5)^2 - 3(-5) + 9 = -25 + 15 + 9 = -1 (which is not greater than zero).

2. When -4.354 < x < 1.354: Substitute x = 0 (for example): -(0)^2 - 3(0) + 9 = 9 (which is greater than zero).

3. When x > 1.354: Substitute x = 2 (for example): -(2)^2 - 3(2) + 9 = -4 (which is not greater than zero).

The inequality is satisfied when -4.354 < x < 1.354. Thus, the solution to the inequality is:

-4.354 < x < 1.354