Mister Exam
2/((1/2)*x*sqrt(51))+((1/2)*x*sqrt(5)-2)/((1/2)*x*sqrt(5)-3)-2>=0 inequation
The solution
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x ___
-*\/ 5 - 2
2 2
-------- + ----------- - 2 >= 0
x ____ x ___
-*\/ 51 -*\/ 5 - 3
2 2
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 \geq 0$$
(sqrt(5)*(x/2) - 2)/(sqrt(5)*(x/2) - 3) + 2/((sqrt(51)*(x/2))) - 2 >= 0
Detail solution
Given the inequality:
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 = 0$$
Solve:
Given the equation:
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 = 0$$
Multiply the equation sides by the denominators:
x and -3 + x*sqrt(5)/2
we get:
$$x \left(\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2\right) = 0$$
$$\frac{- \sqrt{5} x^{2} + \frac{4 \sqrt{255} x}{51} + 8 x - \frac{8 \sqrt{51}}{17}}{\sqrt{5} x - 6} = 0$$
$$\frac{- \sqrt{5} x^{2} + \frac{4 \sqrt{255} x}{51} + 8 x - \frac{8 \sqrt{51}}{17}}{\sqrt{5} x - 6} \left(\frac{\sqrt{5} x}{2} - 3\right) = 0 \left(\frac{\sqrt{5} x}{2} - 3\right)$$
$$- \frac{\sqrt{5} x^{2}}{2} + \frac{2 \sqrt{255} x}{51} + 4 x - \frac{4 \sqrt{51}}{17} = 0$$
This equation is of the form
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
$$a = - \frac{\sqrt{5}}{2}$$
$$b = \frac{2 \sqrt{255}}{51} + 4$$
$$c = - \frac{4 \sqrt{51}}{17}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
This roots
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
=
$$- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
substitute to the expression
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 \geq 0$$
$$-2 + \left(\frac{-2 + \sqrt{5} \frac{- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}}{2}}{-3 + \sqrt{5} \frac{- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}}{2}} + \frac{2}{\sqrt{51} \frac{- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}}{2}}\right) \geq 0$$
one of the solutions of our inequality is:
$$x \leq - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x \geq - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 = 0$$
Solve:
Given the equation:
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 = 0$$
Multiply the equation sides by the denominators:
x and -3 + x*sqrt(5)/2
we get:
$$x \left(\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2\right) = 0$$
$$\frac{- \sqrt{5} x^{2} + \frac{4 \sqrt{255} x}{51} + 8 x - \frac{8 \sqrt{51}}{17}}{\sqrt{5} x - 6} = 0$$
$$\frac{- \sqrt{5} x^{2} + \frac{4 \sqrt{255} x}{51} + 8 x - \frac{8 \sqrt{51}}{17}}{\sqrt{5} x - 6} \left(\frac{\sqrt{5} x}{2} - 3\right) = 0 \left(\frac{\sqrt{5} x}{2} - 3\right)$$
$$- \frac{\sqrt{5} x^{2}}{2} + \frac{2 \sqrt{255} x}{51} + 4 x - \frac{4 \sqrt{51}}{17} = 0$$
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
$$a = - \frac{\sqrt{5}}{2}$$
$$b = \frac{2 \sqrt{255}}{51} + 4$$
$$c = - \frac{4 \sqrt{51}}{17}$$
, then
D = b^2 - 4 * a * c =
(4 + 2*sqrt(255)/51)^2 - 4 * (-sqrt(5)/2) * (-4*sqrt(51)/17) = (4 + 2*sqrt(255)/51)^2 - 8*sqrt(255)/17
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
This roots
$$x_{1} = - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x_{2} = - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
=
$$- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
substitute to the expression
$$\left(\frac{\sqrt{5} \frac{x}{2} - 2}{\sqrt{5} \frac{x}{2} - 3} + \frac{2}{\sqrt{51} \frac{x}{2}}\right) - 2 \geq 0$$
$$-2 + \left(\frac{-2 + \sqrt{5} \frac{- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}}{2}}{-3 + \sqrt{5} \frac{- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}}{2}} + \frac{2}{\sqrt{51} \frac{- \frac{1}{10} - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}}{2}}\right) \geq 0$$
/ / ______________________________ \\
| | / 2 ||
| | / / _____\ _____ _____||
| ___ | / | 2*\/ 255 | 8*\/ 255 2*\/ 255 ||
| \/ 5 *|-4 + / |4 + ---------| - --------- - ---------||
___ | 1 \ \/ \ 51 / 17 51 /|
-2 + \/ 5 *|- -- - -------------------------------------------------------------| ____
\ 20 10 / 2*\/ 51
-2 + --------------------------------------------------------------------------------- + ------------------------------------------------------------------------- >= 0
/ / ______________________________ \\ / / ______________________________ \\
| | / 2 || | | / 2 ||
| | / / _____\ _____ _____|| | | / / _____\ _____ _____||
| ___ | / | 2*\/ 255 | 8*\/ 255 2*\/ 255 || | ___ | / | 2*\/ 255 | 8*\/ 255 2*\/ 255 ||
| \/ 5 *|-4 + / |4 + ---------| - --------- - ---------|| | \/ 5 *|-4 + / |4 + ---------| - --------- - ---------||
___ | 1 \ \/ \ 51 / 17 51 /| | 1 \ \/ \ 51 / 17 51 /|
-3 + \/ 5 *|- -- - -------------------------------------------------------------| 51*|- -- - -------------------------------------------------------------|
\ 20 10 / \ 20 10 / one of the solutions of our inequality is:
$$x \leq - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
_____ _____
\ /
-------•-------•-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq - \frac{\sqrt{5} \left(-4 - \frac{2 \sqrt{255}}{51} + \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}}\right)}{5}$$
$$x \geq - \frac{\sqrt{5} \left(-4 - \sqrt{- \frac{8 \sqrt{255}}{17} + \left(\frac{2 \sqrt{255}}{51} + 4\right)^{2}} - \frac{2 \sqrt{255}}{51}\right)}{5}$$
Solving inequality on a graph
Rapid solution
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/ / _________________ \ / _________________ \\ | | _____ / _____ ___ / _____\ | | ___ / _____\ _____ / _____ ___ || | | 2*\/ 255 *\/ 209 - 2*\/ 255 2*\/ 5 *\102 + \/ 255 / | | 2*\/ 5 *\102 + \/ 255 / 2*\/ 255 *\/ 209 - 2*\/ 255 6*\/ 5 || Or|And|x <= - ------------------------------ + -----------------------, 0 < x|, And|x <= ----------------------- + ------------------------------, ------- < x|| \ \ 255 255 / \ 255 255 5 //
$$\left(x \leq - \frac{2 \sqrt{255} \sqrt{209 - 2 \sqrt{255}}}{255} + \frac{2 \sqrt{5} \left(\sqrt{255} + 102\right)}{255} \wedge 0 < x\right) \vee \left(x \leq \frac{2 \sqrt{255} \sqrt{209 - 2 \sqrt{255}}}{255} + \frac{2 \sqrt{5} \left(\sqrt{255} + 102\right)}{255} \wedge \frac{6 \sqrt{5}}{5} < x\right)$$
((0 < x)∧(x <= -2*sqrt(255)*sqrt(209 - 2*sqrt(255))/255 + 2*sqrt(5)*(102 + sqrt(255))/255))∨((6*sqrt(5)/5 < x)∧(x <= 2*sqrt(5)*(102 + sqrt(255))/255 + 2*sqrt(255)*sqrt(209 - 2*sqrt(255))/255))
Rapid solution 2
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_________________ _________________
_____ / _____ ___ / _____\ ___ ___ / _____\ _____ / _____
2*\/ 255 *\/ 209 - 2*\/ 255 2*\/ 5 *\102 + \/ 255 / 6*\/ 5 2*\/ 5 *\102 + \/ 255 / 2*\/ 255 *\/ 209 - 2*\/ 255
(0, - ------------------------------ + -----------------------] U (-------, ----------------------- + ------------------------------]
255 255 5 255 255
$$x\ in\ \left(0, - \frac{2 \sqrt{255} \sqrt{209 - 2 \sqrt{255}}}{255} + \frac{2 \sqrt{5} \left(\sqrt{255} + 102\right)}{255}\right] \cup \left(\frac{6 \sqrt{5}}{5}, \frac{2 \sqrt{255} \sqrt{209 - 2 \sqrt{255}}}{255} + \frac{2 \sqrt{5} \left(\sqrt{255} + 102\right)}{255}\right]$$
x in Union(Interval.Lopen(0, -2*sqrt(255)*sqrt(209 - 2*sqrt(255))/255 + 2*sqrt(5)*(sqrt(255) + 102)/255), Interval.Lopen(6*sqrt(5)/5, 2*sqrt(255)*sqrt(209 - 2*sqrt(255))/255 + 2*sqrt(5)*(sqrt(255) + 102)/255))