2*sqrt(x-3)-sqrt(x+2)>=11 inequation

Given the inequality:
$$2 \sqrt{x - 3} - \sqrt{x + 2} \geq 11$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \sqrt{x - 3} - \sqrt{x + 2} = 11$$
Solve:
Given the equation
$$2 \sqrt{x - 3} - \sqrt{x + 2} = 11$$
We raise the equation sides to 2-th degree
$$\left(2 \sqrt{x - 3} - \sqrt{x + 2}\right)^{2} = 121$$
or
$$\left(-1\right)^{2} \left(x + 2\right) + \left(\left(-1\right) 2 \cdot 2 \sqrt{\left(x - 3\right) \left(x + 2\right)} + 2^{2} \left(x - 3\right)\right) = 121$$
or
$$5 x - 4 \sqrt{x^{2} - x - 6} - 10 = 121$$
transform:
$$- 4 \sqrt{x^{2} - x - 6} = 131 - 5 x$$
We raise the equation sides to 2-th degree
$$16 x^{2} - 16 x - 96 = \left(131 - 5 x\right)^{2}$$
$$16 x^{2} - 16 x - 96 = 25 x^{2} - 1310 x + 17161$$
Transfer the right side of the equation left part with negative sign
$$- 9 x^{2} + 1294 x - 17257 = 0$$
This equation is of the form

a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -9$$
$$b = 1294$$
$$c = -17257$$
, then

D = b^2 - 4 * a * c = 

(1294)^2 - 4 * (-9) * (-17257) = 1053184

Because D > 0, then the equation has two roots.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{647}{9} - \frac{88 \sqrt{34}}{9}$$
$$x_{2} = \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$

Because
$$\sqrt{x^{2} - x - 6} = \frac{5 x}{4} - \frac{131}{4}$$
and
$$\sqrt{x^{2} - x - 6} \geq 0$$
then
$$\frac{5 x}{4} - \frac{131}{4} \geq 0$$
or
$$\frac{131}{5} \leq x$$
$$x < \infty$$
$$x_{2} = \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$
check:
$$x_{1} = \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$
$$2 \sqrt{x_{1} - 3} - \sqrt{x_{1} + 2} - 11 = 0$$
=
$$-11 + \left(- \sqrt{2 + \left(\frac{88 \sqrt{34}}{9} + \frac{647}{9}\right)} + 2 \sqrt{-3 + \left(\frac{88 \sqrt{34}}{9} + \frac{647}{9}\right)}\right) = 0$$
=

0 = 0

- the identity
The final answer:
$$x_{1} = \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$
$$x_{1} = \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$
$$x_{1} = \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$
This roots
$$x_{1} = \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{88 \sqrt{34}}{9} + \frac{647}{9}\right)$$
=
$$\frac{88 \sqrt{34}}{9} + \frac{6461}{90}$$
substitute to the expression
$$2 \sqrt{x - 3} - \sqrt{x + 2} \geq 11$$
$$- \sqrt{2 + \left(\frac{88 \sqrt{34}}{9} + \frac{6461}{90}\right)} + 2 \sqrt{-3 + \left(\frac{88 \sqrt{34}}{9} + \frac{6461}{90}\right)} \geq 11$$

       __________________          __________________      
      /             ____          /             ____       
     /  6641   88*\/ 34          /  6191   88*\/ 34   >= 11
-   /   ---- + ---------  + 2*  /   ---- + ---------       
  \/     90        9          \/     90        9           

but

       __________________          __________________     
      /             ____          /             ____      
     /  6641   88*\/ 34          /  6191   88*\/ 34   < 11
-   /   ---- + ---------  + 2*  /   ---- + ---------      
  \/     90        9          \/     90        9          

Then
$$x \leq \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$
no execute
the solution of our inequality is:
$$x \geq \frac{88 \sqrt{34}}{9} + \frac{647}{9}$$

         _____  
        /
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       x1