Mister Exam
(5-6x)/(2x+5)<1 inequation
The solution
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5 - 6*x ------- < 1 2*x + 5
$$\frac{5 - 6 x}{2 x + 5} < 1$$
(5 - 6*x)/(2*x + 5) < 1
Detail solution
Given the inequality:
$$\frac{5 - 6 x}{2 x + 5} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{5 - 6 x}{2 x + 5} = 1$$
Solve:
Given the equation:
$$\frac{5 - 6 x}{2 x + 5} = 1$$
Multiply the equation sides by the denominator 5 + 2*x
we get:
$$5 - 6 x = 2 x + 5$$
Move free summands (without x)
from left part to right part, we given:
$$- 6 x = 2 x$$
Move the summands with the unknown x
from the right part to the left part:
$$\left(-8\right) x = 0$$
Divide both parts of the equation by -8
$$x_{1} = 0$$
$$x_{1} = 0$$
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
$$\frac{5 - 6 x}{2 x + 5} < 1$$
$$\frac{5 - \frac{\left(-1\right) 6}{10}}{\frac{\left(-1\right) 2}{10} + 5} < 1$$
but
Then
$$x < 0$$
no execute
the solution of our inequality is:
$$x > 0$$
$$\frac{5 - 6 x}{2 x + 5} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{5 - 6 x}{2 x + 5} = 1$$
Solve:
Given the equation:
$$\frac{5 - 6 x}{2 x + 5} = 1$$
Multiply the equation sides by the denominator 5 + 2*x
we get:
$$5 - 6 x = 2 x + 5$$
Move free summands (without x)
from left part to right part, we given:
$$- 6 x = 2 x$$
Move the summands with the unknown x
from the right part to the left part:
$$\left(-8\right) x = 0$$
Divide both parts of the equation by -8
x = 0 / (-8)
$$x_{1} = 0$$
$$x_{1} = 0$$
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
$$\frac{5 - 6 x}{2 x + 5} < 1$$
$$\frac{5 - \frac{\left(-1\right) 6}{10}}{\frac{\left(-1\right) 2}{10} + 5} < 1$$
7/6 < 1
but
7/6 > 1
Then
$$x < 0$$
no execute
the solution of our inequality is:
$$x > 0$$
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Solving inequality on a graph
Rapid solution 2
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(-oo, -5/2) U (0, oo)
$$x\ in\ \left(-\infty, - \frac{5}{2}\right) \cup \left(0, \infty\right)$$
x in Union(Interval.open(-oo, -5/2), Interval.open(0, oo))
Rapid solution
[src]
Or(And(-oo < x, x < -5/2), And(0 < x, x < oo))
$$\left(-\infty < x \wedge x < - \frac{5}{2}\right) \vee \left(0 < x \wedge x < \infty\right)$$
((-oo < x)∧(x < -5/2))∨((0 < x)∧(x < oo))