Mister Exam
5x^3-35x+30>=0 inequation
The solution
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3 5*x - 35*x + 30 >= 0
$$\left(5 x^{3} - 35 x\right) + 30 \geq 0$$
5*x^3 - 35*x + 30 >= 0
Detail solution
Given the inequality:
$$\left(5 x^{3} - 35 x\right) + 30 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(5 x^{3} - 35 x\right) + 30 = 0$$
Solve:
$$x_{1} = 1$$
$$x_{2} = 2$$
$$x_{3} = -3$$
$$x_{1} = 1$$
$$x_{2} = 2$$
$$x_{3} = -3$$
This roots
$$x_{3} = -3$$
$$x_{1} = 1$$
$$x_{2} = 2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{3}$$
For example, let's take the point
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$-3 + - \frac{1}{10}$$
=
$$- \frac{31}{10}$$
substitute to the expression
$$\left(5 x^{3} - 35 x\right) + 30 \geq 0$$
$$\left(5 \left(- \frac{31}{10}\right)^{3} - \frac{\left(-31\right) 35}{10}\right) + 30 \geq 0$$
but
Then
$$x \leq -3$$
no execute
one of the solutions of our inequality is:
$$x \geq -3 \wedge x \leq 1$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -3 \wedge x \leq 1$$
$$x \geq 2$$
$$\left(5 x^{3} - 35 x\right) + 30 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(5 x^{3} - 35 x\right) + 30 = 0$$
Solve:
$$x_{1} = 1$$
$$x_{2} = 2$$
$$x_{3} = -3$$
$$x_{1} = 1$$
$$x_{2} = 2$$
$$x_{3} = -3$$
This roots
$$x_{3} = -3$$
$$x_{1} = 1$$
$$x_{2} = 2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{3}$$
For example, let's take the point
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$-3 + - \frac{1}{10}$$
=
$$- \frac{31}{10}$$
substitute to the expression
$$\left(5 x^{3} - 35 x\right) + 30 \geq 0$$
$$\left(5 \left(- \frac{31}{10}\right)^{3} - \frac{\left(-31\right) 35}{10}\right) + 30 \geq 0$$
-2091 ------ >= 0 200
but
-2091 ------ < 0 200
Then
$$x \leq -3$$
no execute
one of the solutions of our inequality is:
$$x \geq -3 \wedge x \leq 1$$
_____ _____
/ \ /
-------•-------•-------•-------
x3 x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -3 \wedge x \leq 1$$
$$x \geq 2$$
Rapid solution
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Or(And(-3 <= x, x <= 1), And(2 <= x, x < oo))
$$\left(-3 \leq x \wedge x \leq 1\right) \vee \left(2 \leq x \wedge x < \infty\right)$$
((-3 <= x)∧(x <= 1))∨((2 <= x)∧(x < oo))
Rapid solution 2
[src]
[-3, 1] U [2, oo)
$$x\ in\ \left[-3, 1\right] \cup \left[2, \infty\right)$$
x in Union(Interval(-3, 1), Interval(2, oo))