Mister Exam
(x^2-10*x+17)*(x^2-10*x-25)<=-41 inequation
The solution
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/ 2 \ / 2 \ \x - 10*x + 17/*\x - 10*x - 25/ <= -41
$$\left(\left(x^{2} - 10 x\right) - 25\right) \left(\left(x^{2} - 10 x\right) + 17\right) \leq -41$$
(x^2 - 10*x - 25)*(x^2 - 10*x + 17) <= -41
Detail solution
Given the inequality:
$$\left(\left(x^{2} - 10 x\right) - 25\right) \left(\left(x^{2} - 10 x\right) + 17\right) \leq -41$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} - 10 x\right) - 25\right) \left(\left(x^{2} - 10 x\right) + 17\right) = -41$$
Solve:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{3} = 8$$
$$x_{4} = 12$$
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{3} = 8$$
$$x_{4} = 12$$
This roots
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{3} = 8$$
$$x_{4} = 12$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\left(\left(x^{2} - 10 x\right) - 25\right) \left(\left(x^{2} - 10 x\right) + 17\right) \leq -41$$
$$\left(-25 + \left(\left(- \frac{21}{10}\right)^{2} - \frac{\left(-21\right) 10}{10}\right)\right) \left(17 + \left(\left(- \frac{21}{10}\right)^{2} - \frac{\left(-21\right) 10}{10}\right)\right) \leq -41$$
but
Then
$$x \leq -2$$
no execute
one of the solutions of our inequality is:
$$x \geq -2 \wedge x \leq 2$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -2 \wedge x \leq 2$$
$$x \geq 8 \wedge x \leq 12$$
$$\left(\left(x^{2} - 10 x\right) - 25\right) \left(\left(x^{2} - 10 x\right) + 17\right) \leq -41$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} - 10 x\right) - 25\right) \left(\left(x^{2} - 10 x\right) + 17\right) = -41$$
Solve:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{3} = 8$$
$$x_{4} = 12$$
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{3} = 8$$
$$x_{4} = 12$$
This roots
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{3} = 8$$
$$x_{4} = 12$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\left(\left(x^{2} - 10 x\right) - 25\right) \left(\left(x^{2} - 10 x\right) + 17\right) \leq -41$$
$$\left(-25 + \left(\left(- \frac{21}{10}\right)^{2} - \frac{\left(-21\right) 10}{10}\right)\right) \left(17 + \left(\left(- \frac{21}{10}\right)^{2} - \frac{\left(-21\right) 10}{10}\right)\right) \leq -41$$
173881 ------ <= -41 10000
but
173881 ------ >= -41 10000
Then
$$x \leq -2$$
no execute
one of the solutions of our inequality is:
$$x \geq -2 \wedge x \leq 2$$
_____ _____
/ \ / \
-------•-------•-------•-------•-------
x1 x2 x3 x4Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -2 \wedge x \leq 2$$
$$x \geq 8 \wedge x \leq 12$$
Solving inequality on a graph
Rapid solution
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Or(And(-2 <= x, x <= 2), And(8 <= x, x <= 12))
$$\left(-2 \leq x \wedge x \leq 2\right) \vee \left(8 \leq x \wedge x \leq 12\right)$$
((-2 <= x)∧(x <= 2))∨((8 <= x)∧(x <= 12))
Rapid solution 2
[src]
[-2, 2] U [8, 12]
$$x\ in\ \left[-2, 2\right] \cup \left[8, 12\right]$$
x in Union(Interval(-2, 2), Interval(8, 12))