Mister Exam
(3x+2)/(5x-1)<0 inequation
The solution
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3*x + 2 ------- < 0 5*x - 1
$$\frac{3 x + 2}{5 x - 1} < 0$$
(3*x + 2)/(5*x - 1) < 0
Detail solution
Given the inequality:
$$\frac{3 x + 2}{5 x - 1} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 x + 2}{5 x - 1} = 0$$
Solve:
Given the equation:
$$\frac{3 x + 2}{5 x - 1} = 0$$
Multiply the equation sides by the denominator -1 + 5*x
we get:
$$3 x + 2 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$3 x = -2$$
Divide both parts of the equation by 3
$$x_{1} = - \frac{2}{3}$$
$$x_{1} = - \frac{2}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{2}{3} + - \frac{1}{10}$$
=
$$- \frac{23}{30}$$
substitute to the expression
$$\frac{3 x + 2}{5 x - 1} < 0$$
$$\frac{\frac{\left(-23\right) 3}{30} + 2}{\frac{\left(-23\right) 5}{30} - 1} < 0$$
but
Then
$$x < - \frac{2}{3}$$
no execute
the solution of our inequality is:
$$x > - \frac{2}{3}$$
$$\frac{3 x + 2}{5 x - 1} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 x + 2}{5 x - 1} = 0$$
Solve:
Given the equation:
$$\frac{3 x + 2}{5 x - 1} = 0$$
Multiply the equation sides by the denominator -1 + 5*x
we get:
$$3 x + 2 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$3 x = -2$$
Divide both parts of the equation by 3
x = -2 / (3)
$$x_{1} = - \frac{2}{3}$$
$$x_{1} = - \frac{2}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{2}{3} + - \frac{1}{10}$$
=
$$- \frac{23}{30}$$
substitute to the expression
$$\frac{3 x + 2}{5 x - 1} < 0$$
$$\frac{\frac{\left(-23\right) 3}{30} + 2}{\frac{\left(-23\right) 5}{30} - 1} < 0$$
9/145 < 0
but
9/145 > 0
Then
$$x < - \frac{2}{3}$$
no execute
the solution of our inequality is:
$$x > - \frac{2}{3}$$
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Solving inequality on a graph
Rapid solution
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And(-2/3 < x, x < 1/5)
$$- \frac{2}{3} < x \wedge x < \frac{1}{5}$$
(-2/3 < x)∧(x < 1/5)
Rapid solution 2
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(-2/3, 1/5)
$$x\ in\ \left(- \frac{2}{3}, \frac{1}{5}\right)$$
x in Interval.open(-2/3, 1/5)