Mister Exam
-3*x^2+4<0 inequation
The solution
Detail solution
Given the inequality:
$$4 - 3 x^{2} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$4 - 3 x^{2} = 0$$
Solve:
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
$$a = -3$$
$$b = 0$$
$$c = 4$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
This roots
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{2 \sqrt{3}}{3} - \frac{1}{10}$$
=
$$- \frac{2 \sqrt{3}}{3} - \frac{1}{10}$$
substitute to the expression
$$4 - 3 x^{2} < 0$$
$$4 - 3 \left(- \frac{2 \sqrt{3}}{3} - \frac{1}{10}\right)^{2} < 0$$
one of the solutions of our inequality is:
$$x < - \frac{2 \sqrt{3}}{3}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < - \frac{2 \sqrt{3}}{3}$$
$$x > \frac{2 \sqrt{3}}{3}$$
$$4 - 3 x^{2} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$4 - 3 x^{2} = 0$$
Solve:
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
$$a = -3$$
$$b = 0$$
$$c = 4$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-3) * (4) = 48
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
This roots
$$x_{1} = - \frac{2 \sqrt{3}}{3}$$
$$x_{2} = \frac{2 \sqrt{3}}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{2 \sqrt{3}}{3} - \frac{1}{10}$$
=
$$- \frac{2 \sqrt{3}}{3} - \frac{1}{10}$$
substitute to the expression
$$4 - 3 x^{2} < 0$$
$$4 - 3 \left(- \frac{2 \sqrt{3}}{3} - \frac{1}{10}\right)^{2} < 0$$
2
/ ___\
| 1 2*\/ 3 | < 0
4 - 3*|- -- - -------|
\ 10 3 / one of the solutions of our inequality is:
$$x < - \frac{2 \sqrt{3}}{3}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < - \frac{2 \sqrt{3}}{3}$$
$$x > \frac{2 \sqrt{3}}{3}$$
Solving inequality on a graph
Rapid solution 2
[src]
___ ___
-2*\/ 3 2*\/ 3
(-oo, --------) U (-------, oo)
3 3
$$x\ in\ \left(-\infty, - \frac{2 \sqrt{3}}{3}\right) \cup \left(\frac{2 \sqrt{3}}{3}, \infty\right)$$
x in Union(Interval.open(-oo, -2*sqrt(3)/3), Interval.open(2*sqrt(3)/3, oo))
Rapid solution
[src]
/ / ___\ / ___ \\ | | -2*\/ 3 | |2*\/ 3 || Or|And|-oo < x, x < --------|, And|------- < x, x < oo|| \ \ 3 / \ 3 //
$$\left(-\infty < x \wedge x < - \frac{2 \sqrt{3}}{3}\right) \vee \left(\frac{2 \sqrt{3}}{3} < x \wedge x < \infty\right)$$
((-oo < x)∧(x < -2*sqrt(3)/3))∨((x < oo)∧(2*sqrt(3)/3 < x))