Question

√х^2+3х-18>2х+3решение неравенств

Answer 1

X+3x-18>2x+3
4x-2x>3+18
2x>21
x>10,5
(10,5;oo)

Answer 2

To solve the inequality √(x^2 + 3x - 18) > 2x + 3, follow these steps:

Step 1: Simplify the expression under the square root. x^2 + 3x - 18 can be factored as (x + 6)(x - 3).

Step 2: Rewrite the inequality. √(x^2 + 3x - 18) > 2x + 3

Step 3: Set up two separate cases for the square root: Case 1: √(x^2 + 3x - 18) > 2x + 3 Case 2: √(x^2 + 3x - 18) < -(2x + 3)

Let's solve each case individually:

Case 1: √(x^2 + 3x - 18) > 2x + 3

Square both sides to eliminate the square root (remember to consider the non-negativity condition of the square root): x^2 + 3x - 18 > (2x + 3)^2

Expand and simplify: x^2 + 3x - 18 > 4x^2 + 12x + 9

Move all terms to one side of the inequality: 0 > 3x^2 + 9x + 27

Divide by 3 to simplify: 0 > x^2 + 3x + 9

At this point, it's clear that the quadratic equation on the right side has no real solutions (since the discriminant is negative). Thus, Case 1 has no valid solution.

Case 2: √(x^2 + 3x - 18) < -(2x + 3)

Square both sides (remember to reverse the inequality since we are squaring a negative number): x^2 + 3x - 18 < (2x + 3)^2

Expand and simplify: x^2 + 3x - 18 < 4x^2 + 12x + 9

Move all terms to one side of the inequality: 0 < 3x^2 + 9x + 27

Divide by 3 to simplify: 0 < x^2 + 3x + 9

Now, we have another quadratic equation on the right side. To find its solutions, we can use the discriminant. The discriminant (Δ) is given by Δ = b^2 - 4ac, where the quadratic equation is ax^2 + bx + c = 0.

In our case, a = 1, b = 3, and c = 9. Δ = 3^2 - 4 * 1 * 9 = 9 - 36 = -27

Since the discriminant is negative (Δ < 0), the quadratic equation has no real solutions. Therefore, Case 2 also has no valid solution.

Overall, the original inequality √(x^2 + 3x - 18) > 2x + 3 has no real solutions, meaning there are no values of x that satisfy the inequality.