Mister Exam
(3*x+1)/(x+2)>0 inequation
The solution
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3*x + 1 ------- > 0 x + 2
$$\frac{3 x + 1}{x + 2} > 0$$
(3*x + 1)/(x + 2) > 0
Detail solution
Given the inequality:
$$\frac{3 x + 1}{x + 2} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 x + 1}{x + 2} = 0$$
Solve:
Given the equation:
$$\frac{3 x + 1}{x + 2} = 0$$
Multiply the equation sides by the denominator 2 + x
we get:
$$3 x + 1 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$3 x = -1$$
Divide both parts of the equation by 3
$$x_{1} = - \frac{1}{3}$$
$$x_{1} = - \frac{1}{3}$$
This roots
$$x_{1} = - \frac{1}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{3} + - \frac{1}{10}$$
=
$$- \frac{13}{30}$$
substitute to the expression
$$\frac{3 x + 1}{x + 2} > 0$$
$$\frac{\frac{\left(-13\right) 3}{30} + 1}{- \frac{13}{30} + 2} > 0$$
Then
$$x < - \frac{1}{3}$$
no execute
the solution of our inequality is:
$$x > - \frac{1}{3}$$
$$\frac{3 x + 1}{x + 2} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 x + 1}{x + 2} = 0$$
Solve:
Given the equation:
$$\frac{3 x + 1}{x + 2} = 0$$
Multiply the equation sides by the denominator 2 + x
we get:
$$3 x + 1 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$3 x = -1$$
Divide both parts of the equation by 3
x = -1 / (3)
$$x_{1} = - \frac{1}{3}$$
$$x_{1} = - \frac{1}{3}$$
This roots
$$x_{1} = - \frac{1}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{3} + - \frac{1}{10}$$
=
$$- \frac{13}{30}$$
substitute to the expression
$$\frac{3 x + 1}{x + 2} > 0$$
$$\frac{\frac{\left(-13\right) 3}{30} + 1}{- \frac{13}{30} + 2} > 0$$
-9/47 > 0
Then
$$x < - \frac{1}{3}$$
no execute
the solution of our inequality is:
$$x > - \frac{1}{3}$$
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Solving inequality on a graph
Rapid solution
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Or(And(-oo < x, x < -2), And(-1/3 < x, x < oo))
$$\left(-\infty < x \wedge x < -2\right) \vee \left(- \frac{1}{3} < x \wedge x < \infty\right)$$
((-oo < x)∧(x < -2))∨((-1/3 < x)∧(x < oo))
Rapid solution 2
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(-oo, -2) U (-1/3, oo)
$$x\ in\ \left(-\infty, -2\right) \cup \left(- \frac{1}{3}, \infty\right)$$
x in Union(Interval.open(-oo, -2), Interval.open(-1/3, oo))