Mister Exam
(x+8*x)/(x-4)>0 inequation
The solution
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x + 8*x ------- > 0 x - 4
$$\frac{x + 8 x}{x - 4} > 0$$
(x + 8*x)/(x - 4) > 0
Detail solution
Given the inequality:
$$\frac{x + 8 x}{x - 4} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x + 8 x}{x - 4} = 0$$
Solve:
Given the equation:
$$\frac{x + 8 x}{x - 4} = 0$$
Multiply the equation sides by the denominator -4 + x
we get:
$$9 x = 0$$
Divide both parts of the equation by 9
$$x_{1} = 0$$
$$x_{1} = 0$$
This roots
$$x_{1} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\frac{x + 8 x}{x - 4} > 0$$
$$\frac{\frac{\left(-1\right) 8}{10} - \frac{1}{10}}{-4 - \frac{1}{10}} > 0$$
the solution of our inequality is:
$$x < 0$$
$$\frac{x + 8 x}{x - 4} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x + 8 x}{x - 4} = 0$$
Solve:
Given the equation:
$$\frac{x + 8 x}{x - 4} = 0$$
Multiply the equation sides by the denominator -4 + x
we get:
$$9 x = 0$$
Divide both parts of the equation by 9
x = 0 / (9)
$$x_{1} = 0$$
$$x_{1} = 0$$
This roots
$$x_{1} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\frac{x + 8 x}{x - 4} > 0$$
$$\frac{\frac{\left(-1\right) 8}{10} - \frac{1}{10}}{-4 - \frac{1}{10}} > 0$$
9/41 > 0
the solution of our inequality is:
$$x < 0$$
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Solving inequality on a graph
Rapid solution
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Or(And(-oo < x, x < 0), And(4 < x, x < oo))
$$\left(-\infty < x \wedge x < 0\right) \vee \left(4 < x \wedge x < \infty\right)$$
((-oo < x)∧(x < 0))∨((4 < x)∧(x < oo))
Rapid solution 2
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(-oo, 0) U (4, oo)
$$x\ in\ \left(-\infty, 0\right) \cup \left(4, \infty\right)$$
x in Union(Interval.open(-oo, 0), Interval.open(4, oo))