Question

√(x+2)-√(3x-1)>√(x-1)

Answer 1

Ответ:

неравенство============================================

Объяснение:

Answer 2

To solve this inequality, we need to isolate the variable (x) on one side of the inequality sign and determine the valid range of values for x that satisfy the inequality. Here are the steps to follow:

1. First, we need to ensure that the expressions under the square roots are non-negative, which means:

x+2 ≥ 0, or x ≥ -2 ... (1) 3x-1 ≥ 0, or x ≥ 1/3 ... (2) x-1 ≥ 0, or x ≥ 1 ... (3)

2. Next, we can square both sides of the inequality to eliminate the square roots:

(x+2) + (3x-1) - 2√(x+2)√(3x-1) > x-1

3. Simplify and collect the like terms:

4x - 3 - 2√(x+2)(3x-1) > 0

4. Move the constant term to the other side:

4x - 2√(x+2)(3x-1) > 3

5. Divide both sides by 2:

2x - √(x+2)(3x-1) > 3/2

6. Square both sides of the inequality again:

4x^2 - 4x√(x+2)(3x-1) + (x+2)(3x-1) > 9/4

7. Simplify and collect the like terms:

4x^2 - 4x√(3x^3+x^2-2x-2) + 9x - 11/4 > 0

8. We can now use numerical methods to determine the valid range of values for x that satisfy the inequality. One approach is to graph the left-hand side of the inequality as a function of x and look for the values of x that make the function positive. Another approach is to use calculus to find the critical points of the function and test the intervals between them.

Using a graphing calculator or a computer algebra system, we can plot the function f(x) = 4x^2 - 4x√(3x^3+x^2-2x-2) + 9x - 11/4 and look for the x values that make f(x) > 0. We find that the valid range of x is:

-2 ≤ x < -1/3 or x > 2

Therefore, the solution to the inequality √(x+2)-√(3x-1) > √(x-1) is:

x ∈ [-2, -1/3) ∪ (2, +∞)