Mister Exam
(1.1-√2)(1-x^2)<0 inequation
The solution
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/11 ___\ / 2\ |-- - \/ 2 |*\1 - x / < 0 \10 /
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) < 0$$
(1 - x^2)*(11/10 - sqrt(2)) < 0
Detail solution
Given the inequality:
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) = 0$$
Solve:
Expand the expression in the equation
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) = 0$$
We get the quadratic equation
$$- \frac{11 x^{2}}{10} + \sqrt{2} x^{2} - \sqrt{2} + \frac{11}{10} = 0$$
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
$$a = - \frac{11}{10} + \sqrt{2}$$
$$b = 0$$
$$c = \frac{11}{10} - \sqrt{2}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
This roots
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} + - \frac{1}{10}$$
=
$$- \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} - \frac{1}{10}$$
substitute to the expression
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) < 0$$
$$\left(1 - \left(- \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} - \frac{1}{10}\right)^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) < 0$$
but
Then
$$x < - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} \wedge x < \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) = 0$$
Solve:
Expand the expression in the equation
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) = 0$$
We get the quadratic equation
$$- \frac{11 x^{2}}{10} + \sqrt{2} x^{2} - \sqrt{2} + \frac{11}{10} = 0$$
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
$$a = - \frac{11}{10} + \sqrt{2}$$
$$b = 0$$
$$c = \frac{11}{10} - \sqrt{2}$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-11/10 + sqrt(2)) * (11/10 - sqrt(2)) = -(-22/5 + 4*sqrt(2))*(11/10 - sqrt(2))
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
This roots
$$x_{2} = - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
$$x_{1} = \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} + - \frac{1}{10}$$
=
$$- \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} - \frac{1}{10}$$
substitute to the expression
$$\left(1 - x^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) < 0$$
$$\left(1 - \left(- \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} - \frac{1}{10}\right)^{2}\right) \left(\frac{11}{10} - \sqrt{2}\right) < 0$$
/ 2\ | / ________________ ______________\ | | | / 22 ___ / 11 ___ | | | | / - -- + 4*\/ 2 * / - -- + \/ 2 | | | | 1 \/ 5 \/ 10 | | /11 ___\ < 0 |1 - |- -- - ---------------------------------------| |*|-- - \/ 2 | | | 10 11 ___ | | \10 / | | - -- + 2*\/ 2 | | \ \ 5 / /
but
/ 2\ | / ________________ ______________\ | | | / 22 ___ / 11 ___ | | | | / - -- + 4*\/ 2 * / - -- + \/ 2 | | | | 1 \/ 5 \/ 10 | | /11 ___\ > 0 |1 - |- -- - ---------------------------------------| |*|-- - \/ 2 | | | 10 11 ___ | | \10 / | | - -- + 2*\/ 2 | | \ \ 5 / /
Then
$$x < - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}} \wedge x < \frac{\sqrt{- \frac{22}{5} + 4 \sqrt{2}} \sqrt{- \frac{11}{10} + \sqrt{2}}}{- \frac{11}{5} + 2 \sqrt{2}}$$
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x2 x1
Solving inequality on a graph