Mister Exam
x^2-8x-(3:|x-4|)+18⩽0 inequation
The solution
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2 3
x - 8*x - ------- + 18 <= 0
|x - 4|
$$\left(\left(x^{2} - 8 x\right) - \frac{3}{\left|{x - 4}\right|}\right) + 18 \leq 0$$
x^2 - 8*x - 3/|x - 4| + 18 <= 0
Detail solution
Given the inequality:
$$\left(\left(x^{2} - 8 x\right) - \frac{3}{\left|{x - 4}\right|}\right) + 18 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} - 8 x\right) - \frac{3}{\left|{x - 4}\right|}\right) + 18 = 0$$
Solve:
$$x_{1} = 5$$
$$x_{2} = 3$$
$$x_{1} = 5$$
$$x_{2} = 3$$
This roots
$$x_{2} = 3$$
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$2.9$$
substitute to the expression
$$\left(\left(x^{2} - 8 x\right) - \frac{3}{\left|{x - 4}\right|}\right) + 18 \leq 0$$
$$\left(\left(- 2.9 \cdot 8 + 2.9^{2}\right) - \frac{3}{\left|{-4 + 2.9}\right|}\right) + 18 \leq 0$$
but
Then
$$x \leq 3$$
no execute
one of the solutions of our inequality is:
$$x \geq 3 \wedge x \leq 5$$
$$\left(\left(x^{2} - 8 x\right) - \frac{3}{\left|{x - 4}\right|}\right) + 18 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} - 8 x\right) - \frac{3}{\left|{x - 4}\right|}\right) + 18 = 0$$
Solve:
$$x_{1} = 5$$
$$x_{2} = 3$$
$$x_{1} = 5$$
$$x_{2} = 3$$
This roots
$$x_{2} = 3$$
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$2.9$$
substitute to the expression
$$\left(\left(x^{2} - 8 x\right) - \frac{3}{\left|{x - 4}\right|}\right) + 18 \leq 0$$
$$\left(\left(- 2.9 \cdot 8 + 2.9^{2}\right) - \frac{3}{\left|{-4 + 2.9}\right|}\right) + 18 \leq 0$$
0.482727272727274 <= 0
but
0.482727272727274 >= 0
Then
$$x \leq 3$$
no execute
one of the solutions of our inequality is:
$$x \geq 3 \wedge x \leq 5$$
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Solving inequality on a graph
Rapid solution
[src]
Or(And(3 <= x, x < 4), And(x <= 5, 4 < x))
$$\left(3 \leq x \wedge x < 4\right) \vee \left(x \leq 5 \wedge 4 < x\right)$$
((3 <= x)∧(x < 4))∨((x <= 5)∧(4 < x))
Rapid solution 2
[src]
[3, 4) U (4, 5]
$$x\ in\ \left[3, 4\right) \cup \left(4, 5\right]$$
x in Union(Interval.Ropen(3, 4), Interval.Lopen(4, 5))