Mister Exam
sqrt((3x-2)/(2x+6))>0 inequation
The solution
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_________ / 3*x - 2 / ------- > 0 \/ 2*x + 6
$$\sqrt{\frac{3 x - 2}{2 x + 6}} > 0$$
sqrt((3*x - 2)/(2*x + 6)) > 0
Detail solution
Given the inequality:
$$\sqrt{\frac{3 x - 2}{2 x + 6}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = 0$$
Solve:
$$x_{1} = \frac{2}{3}$$
$$x_{1} = \frac{2}{3}$$
This roots
$$x_{1} = \frac{2}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{2}{3}$$
=
$$\frac{17}{30}$$
substitute to the expression
$$\sqrt{\frac{3 x - 2}{2 x + 6}} > 0$$
$$\sqrt{\frac{-2 + \frac{3 \cdot 17}{30}}{\frac{2 \cdot 17}{30} + 6}} > 0$$
Then
$$x < \frac{2}{3}$$
no execute
the solution of our inequality is:
$$x > \frac{2}{3}$$
$$\sqrt{\frac{3 x - 2}{2 x + 6}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = 0$$
Solve:
$$x_{1} = \frac{2}{3}$$
$$x_{1} = \frac{2}{3}$$
This roots
$$x_{1} = \frac{2}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{2}{3}$$
=
$$\frac{17}{30}$$
substitute to the expression
$$\sqrt{\frac{3 x - 2}{2 x + 6}} > 0$$
$$\sqrt{\frac{-2 + \frac{3 \cdot 17}{30}}{\frac{2 \cdot 17}{30} + 6}} > 0$$
_____
3*I*\/ 214
----------- > 0
214
Then
$$x < \frac{2}{3}$$
no execute
the solution of our inequality is:
$$x > \frac{2}{3}$$
_____
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x1
Solving inequality on a graph
Rapid solution
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Or(And(-oo < x, x < -3), And(2/3 < x, x < oo))
$$\left(-\infty < x \wedge x < -3\right) \vee \left(\frac{2}{3} < x \wedge x < \infty\right)$$
((-oo < x)∧(x < -3))∨((2/3 < x)∧(x < oo))
Rapid solution 2
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(-oo, -3) U (2/3, oo)
$$x\ in\ \left(-\infty, -3\right) \cup \left(\frac{2}{3}, \infty\right)$$
x in Union(Interval.open(-oo, -3), Interval.open(2/3, oo))