Question

-x(x-7) больше x-4 Помогите,пожалуйста. Спасибо^^

Answer 1

Problem Statement

You are looking for a detailed explanation of the expression -x(x-7) > x-4.

Solution

To solve the inequality -x(x-7) > x-4, we can follow these steps:

1. Simplify the expression on the left side of the inequality. 2. Simplify the expression on the right side of the inequality. 3. Solve the resulting inequality.

Let's go through each step in detail.

Step 1: Simplify the expression on the left side

To simplify the expression -x(x-7), we can use the distributive property of multiplication over addition/subtraction. Applying this property, we get:

-x(x-7) = -x * x + (-x) * (-7) = -x^2 + 7x

So, the left side of the inequality becomes -x^2 + 7x.

Step 2: Simplify the expression on the right side

The expression on the right side of the inequality is already in its simplest form, which is x-4.

Step 3: Solve the inequality

Now that we have simplified both sides of the inequality, we can solve it. The inequality is:

-x^2 + 7x > x - 4

To solve this inequality, we can bring all the terms to one side to form a quadratic equation:

-x^2 + 7x - x + 4 > 0

Simplifying further, we get:

-x^2 + 6x + 4 > 0

To find the solution, we can factorize the quadratic equation or use the quadratic formula. However, in this case, it is easier to use the fact that the coefficient of the x^2 term is negative. This means that the parabola opens downwards, and the inequality will hold true either when the parabola is entirely above the x-axis or when it intersects the x-axis at two distinct points.

To find the x-intercepts of the quadratic equation, we can set it equal to zero:

-x^2 + 6x + 4 = 0

Using the quadratic formula, we can find the solutions:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = -1, b = 6, and c = 4. Plugging in these values, we get:

x = (-6 ± √(6^2 - 4(-1)(4))) / (2(-1))

Simplifying further, we get:

x = (-6 ± √(36 + 16)) / (-2)

x = (-6 ± √52) / (-2)

x = (-6 ± 2√13) / (-2)

Simplifying the expression, we get:

x = 3 ± √13

So, the x-intercepts of the quadratic equation are x = 3 + √13 and x = 3 - √13.

Now, let's analyze the inequality based on the x-intercepts:

- When x < 3 - √13, the quadratic equation is negative, and the inequality is not satisfied. - When 3 - √13 < x < 3 + √13, the quadratic equation is positive, and the inequality is satisfied. - When x > 3 + √13, the quadratic equation is negative, and the inequality is not satisfied.

Therefore, the solution to the inequality -x(x-7) > x-4 is x ∈ (3 - √13, 3 + √13).

Conclusion

The solution to the inequality -x(x-7) > x-4 is x ∈ (3 - √13, 3 + √13). This means that any value of x within the interval (3 - √13, 3 + √13) will satisfy the inequality.