Question

Помогите решить неравенства: a) 5x+4/x<4; б) 6x+1/x+1>1; в) x/x-1>=2; г) 3x-1/x+2>=1

Answer 1

Ну думаю ты всё и сам поймёшь)

Answer 2

a) 5x + 4/x < 4:

To solve this inequality, we'll start by multiplying both sides by x to get rid of the fraction:

5x^2 + 4 < 4x

Next, let's move all terms to one side of the inequality:

5x^2 - 4x + 4 < 0

To solve this quadratic inequality, we can use various methods, such as factoring, completing the square, or the quadratic formula. However, in this case, it seems that the quadratic equation does not factor easily.

Using the quadratic formula, we can find the roots of the quadratic equation:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 5, b = -4, and c = 4. Substituting these values into the formula, we get:

x = (-(-4) ± √((-4)^2 - 4 * 5 * 4)) / (2 * 5) = (4 ± √(16 - 80)) / 10 = (4 ± √(-64)) / 10

Since the discriminant is negative (meaning there are no real solutions), the inequality has no solution.

b) 6x + 1/x + 1 > 1:

Let's start by moving all terms to one side of the inequality:

6x + 1/x + 1 - 1 > 0

Simplifying, we get:

6x + 1/x > 0

Next, we need to find the critical points by setting the expression equal to zero:

6x + 1/x = 0

Multiplying through by x, we get:

6x^2 + 1 = 0

This equation does not have any real solutions since the discriminant is negative.

Now, we can analyze the sign of the expression in different intervals:

For x < 0, both 6x and 1/x are negative, so their sum is negative.

For x > 0, both 6x and 1/x are positive, so their sum is positive.

Therefore, the inequality is satisfied when x < 0 or x > 0. The solution is (-∞, 0) U (0, +∞).

c) x/(x - 1) ≥ 2:

Let's start by moving all terms to one side of the inequality:

x/(x - 1) - 2 ≥ 0

To simplify the expression, let's find a common denominator:

(x - 2(x - 1))/(x - 1) ≥ 0

(-x + 2)/(x - 1) ≥ 0

Next, we'll find the critical points by setting the expression equal to zero:

(-x + 2)/(x - 1) = 0

The critical point is x = 2.

Now, we can analyze the sign of the expression in different intervals:

For x < 0, both -x and (x - 1) are negative, so their quotient is positive.

For 0 < x < 1, -x is negative, and (x - 1) is positive, so their quotient is negative.

For x > 1, both -x and (x - 1) are positive, so their quotient is negative.

Therefore, the inequality is satisfied when x < 0 or x > 1. The solution is (-∞, 0) U (1, +∞).

г) 3x - 1/(x + 2) ≥ 1:

Let's start by moving all terms