Mister Exam
x^2-|5x+8|>0. inequation
The solution
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2 x - |5*x + 8| > 0
$$x^{2} - \left|{5 x + 8}\right| > 0$$
x^2 - |5*x + 8| > 0
Detail solution
Given the inequality:
$$x^{2} - \left|{5 x + 8}\right| > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x^{2} - \left|{5 x + 8}\right| = 0$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.
1.
$$5 x + 8 \geq 0$$
or
$$- \frac{8}{5} \leq x \wedge x < \infty$$
we get the equation
$$x^{2} - \left(5 x + 8\right) = 0$$
after simplifying we get
$$x^{2} - 5 x - 8 = 0$$
the solution in this interval:
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
2.
$$5 x + 8 < 0$$
or
$$-\infty < x \wedge x < - \frac{8}{5}$$
we get the equation
$$x^{2} - \left(- 5 x - 8\right) = 0$$
after simplifying we get
$$x^{2} + 5 x + 8 = 0$$
the solution in this interval:
$$x_{3} = - \frac{5}{2} - \frac{\sqrt{7} i}{2}$$
but x3 not in the inequality interval
$$x_{4} = - \frac{5}{2} + \frac{\sqrt{7} i}{2}$$
but x4 not in the inequality interval
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
This roots
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{5}{2} - \frac{\sqrt{57}}{2}\right) + - \frac{1}{10}$$
=
$$\frac{12}{5} - \frac{\sqrt{57}}{2}$$
substitute to the expression
$$x^{2} - \left|{5 x + 8}\right| > 0$$
$$- \left|{5 \left(\frac{12}{5} - \frac{\sqrt{57}}{2}\right) + 8}\right| + \left(\frac{12}{5} - \frac{\sqrt{57}}{2}\right)^{2} > 0$$
one of the solutions of our inequality is:
$$x < \frac{5}{2} - \frac{\sqrt{57}}{2}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x > \frac{5}{2} + \frac{\sqrt{57}}{2}$$
$$x^{2} - \left|{5 x + 8}\right| > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x^{2} - \left|{5 x + 8}\right| = 0$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.
1.
$$5 x + 8 \geq 0$$
or
$$- \frac{8}{5} \leq x \wedge x < \infty$$
we get the equation
$$x^{2} - \left(5 x + 8\right) = 0$$
after simplifying we get
$$x^{2} - 5 x - 8 = 0$$
the solution in this interval:
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
2.
$$5 x + 8 < 0$$
or
$$-\infty < x \wedge x < - \frac{8}{5}$$
we get the equation
$$x^{2} - \left(- 5 x - 8\right) = 0$$
after simplifying we get
$$x^{2} + 5 x + 8 = 0$$
the solution in this interval:
$$x_{3} = - \frac{5}{2} - \frac{\sqrt{7} i}{2}$$
but x3 not in the inequality interval
$$x_{4} = - \frac{5}{2} + \frac{\sqrt{7} i}{2}$$
but x4 not in the inequality interval
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
This roots
$$x_{1} = \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x_{2} = \frac{5}{2} + \frac{\sqrt{57}}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{5}{2} - \frac{\sqrt{57}}{2}\right) + - \frac{1}{10}$$
=
$$\frac{12}{5} - \frac{\sqrt{57}}{2}$$
substitute to the expression
$$x^{2} - \left|{5 x + 8}\right| > 0$$
$$- \left|{5 \left(\frac{12}{5} - \frac{\sqrt{57}}{2}\right) + 8}\right| + \left(\frac{12}{5} - \frac{\sqrt{57}}{2}\right)^{2} > 0$$
2
/ ____\ ____
|12 \/ 57 | 5*\/ 57 > 0
-20 + |-- - ------| + --------
\5 2 / 2 one of the solutions of our inequality is:
$$x < \frac{5}{2} - \frac{\sqrt{57}}{2}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{5}{2} - \frac{\sqrt{57}}{2}$$
$$x > \frac{5}{2} + \frac{\sqrt{57}}{2}$$
Solving inequality on a graph
Rapid solution
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/ / ____\ / ____ \\ | | 5 \/ 57 | | 5 \/ 57 || Or|And|-oo < x, x < - - ------|, And|x < oo, - + ------ < x|| \ \ 2 2 / \ 2 2 //
$$\left(-\infty < x \wedge x < \frac{5}{2} - \frac{\sqrt{57}}{2}\right) \vee \left(x < \infty \wedge \frac{5}{2} + \frac{\sqrt{57}}{2} < x\right)$$
((-oo < x)∧(x < 5/2 - sqrt(57)/2))∨((x < oo)∧(5/2 + sqrt(57)/2 < x))
Rapid solution 2
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____ ____
5 \/ 57 5 \/ 57
(-oo, - - ------) U (- + ------, oo)
2 2 2 2
$$x\ in\ \left(-\infty, \frac{5}{2} - \frac{\sqrt{57}}{2}\right) \cup \left(\frac{5}{2} + \frac{\sqrt{57}}{2}, \infty\right)$$
x in Union(Interval.open(-oo, 5/2 - sqrt(57)/2), Interval.open(5/2 + sqrt(57)/2, oo))