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6x-2:(x-1)(x+2)<=0 inequation

A inequation with variable

The solution

You have entered [src]
        2               
6*x - -----*(x + 2) <= 0
      x - 1             
$$6 x - \frac{2}{x - 1} \left(x + 2\right) \leq 0$$
6*x - 2/(x - 1)*(x + 2) <= 0
Detail solution
Given the inequality:
$$6 x - \frac{2}{x - 1} \left(x + 2\right) \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$6 x - \frac{2}{x - 1} \left(x + 2\right) = 0$$
Solve:
Given the equation:
$$6 x - \frac{2}{x - 1} \left(x + 2\right) = 0$$
Multiply the equation sides by the denominators:
-1 + x
we get:
$$\left(x - 1\right) \left(6 x - \frac{2}{x - 1} \left(x + 2\right)\right) = 0$$
$$6 x^{2} - 8 x - 4 = 0$$
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
$$a = 6$$
$$b = -8$$
$$c = -4$$
, then
D = b^2 - 4 * a * c = 

(-8)^2 - 4 * (6) * (-4) = 160

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

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

or
$$x_{1} = \frac{2}{3} + \frac{\sqrt{10}}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{10}}{3}$$
$$x_{1} = \frac{2}{3} + \frac{\sqrt{10}}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{10}}{3}$$
$$x_{1} = \frac{2}{3} + \frac{\sqrt{10}}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{10}}{3}$$
This roots
$$x_{2} = \frac{2}{3} - \frac{\sqrt{10}}{3}$$
$$x_{1} = \frac{2}{3} + \frac{\sqrt{10}}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(\frac{2}{3} - \frac{\sqrt{10}}{3}\right) + - \frac{1}{10}$$
=
$$\frac{17}{30} - \frac{\sqrt{10}}{3}$$
substitute to the expression
$$6 x - \frac{2}{x - 1} \left(x + 2\right) \leq 0$$
$$6 \left(\frac{17}{30} - \frac{\sqrt{10}}{3}\right) - \frac{2}{-1 + \left(\frac{17}{30} - \frac{\sqrt{10}}{3}\right)} \left(\left(\frac{17}{30} - \frac{\sqrt{10}}{3}\right) + 2\right) \leq 0$$
                  /       ____\     
                  |77   \/ 10 |     
                2*|-- - ------|     
17       ____     \30     3   /     
-- - 2*\/ 10  - --------------- <= 0
5                         ____      
                   13   \/ 10       
                 - -- - ------      
                   30     3         

one of the solutions of our inequality is:
$$x \leq \frac{2}{3} - \frac{\sqrt{10}}{3}$$
 _____           _____          
      \         /
-------•-------•-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \frac{2}{3} - \frac{\sqrt{10}}{3}$$
$$x \geq \frac{2}{3} + \frac{\sqrt{10}}{3}$$
Solving inequality on a graph
Rapid solution 2 [src]
            ____              ____ 
      2   \/ 10         2   \/ 10  
(-oo, - - ------] U (1, - + ------]
      3     3           3     3    
$$x\ in\ \left(-\infty, \frac{2}{3} - \frac{\sqrt{10}}{3}\right] \cup \left(1, \frac{2}{3} + \frac{\sqrt{10}}{3}\right]$$
x in Union(Interval(-oo, 2/3 - sqrt(10)/3), Interval.Lopen(1, 2/3 + sqrt(10)/3))
Rapid solution [src]
  /   /           ____         \     /           ____       \\
  |   |     2   \/ 10          |     |     2   \/ 10        ||
Or|And|x <= - - ------, -oo < x|, And|x <= - + ------, 1 < x||
  \   \     3     3            /     \     3     3          //
$$\left(x \leq \frac{2}{3} - \frac{\sqrt{10}}{3} \wedge -\infty < x\right) \vee \left(x \leq \frac{2}{3} + \frac{\sqrt{10}}{3} \wedge 1 < x\right)$$
((-oo < x)∧(x <= 2/3 - sqrt(10)/3))∨((1 < x)∧(x <= 2/3 + sqrt(10)/3))