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4x−3x**2−9≤0 inequation

A inequation with variable

The solution

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4*x - 3*x  - 9 <= 0
$$\left(- 3 x^{2} + 4 x\right) - 9 \leq 0$$
-3*x^2 + 4*x - 9 <= 0
Detail solution
Given the inequality:
$$\left(- 3 x^{2} + 4 x\right) - 9 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3 x^{2} + 4 x\right) - 9 = 0$$
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
$$a = -3$$
$$b = 4$$
$$c = -9$$
, then
D = b^2 - 4 * a * c = 

(4)^2 - 4 * (-3) * (-9) = -92

Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)

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

or
$$x_{1} = \frac{2}{3} - \frac{\sqrt{23} i}{3}$$
$$x_{2} = \frac{2}{3} + \frac{\sqrt{23} i}{3}$$
$$x_{1} = \frac{2}{3} - \frac{\sqrt{23} i}{3}$$
$$x_{2} = \frac{2}{3} + \frac{\sqrt{23} i}{3}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0

$$-9 + \left(0 \cdot 4 - 3 \cdot 0^{2}\right) \leq 0$$
-9 <= 0

so the inequality is always executed
Solving inequality on a graph
Rapid solution 2 [src]
(-oo, oo)
$$x\ in\ \left(-\infty, \infty\right)$$
x in Interval(-oo, oo)
Rapid solution [src]
And(-oo < x, x < oo)
$$-\infty < x \wedge x < \infty$$
(-oo < x)∧(x < oo)