Mister Exam
(x-2)/(x-7)>0 inequation
The solution
Detail solution
Given the inequality:
$$\frac{x - 2}{x - 7} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x - 2}{x - 7} = 0$$
Solve:
Given the equation:
$$\frac{x - 2}{x - 7} = 0$$
Multiply the equation sides by the denominator -7 + x
we get:
$$x - 2 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$x = 2$$
$$x_{1} = 2$$
$$x_{1} = 2$$
This roots
$$x_{1} = 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\frac{x - 2}{x - 7} > 0$$
$$\frac{-2 + \frac{19}{10}}{-7 + \frac{19}{10}} > 0$$
the solution of our inequality is:
$$x < 2$$
$$\frac{x - 2}{x - 7} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x - 2}{x - 7} = 0$$
Solve:
Given the equation:
$$\frac{x - 2}{x - 7} = 0$$
Multiply the equation sides by the denominator -7 + x
we get:
$$x - 2 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$x = 2$$
$$x_{1} = 2$$
$$x_{1} = 2$$
This roots
$$x_{1} = 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\frac{x - 2}{x - 7} > 0$$
$$\frac{-2 + \frac{19}{10}}{-7 + \frac{19}{10}} > 0$$
1/51 > 0
the solution of our inequality is:
$$x < 2$$
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Solving inequality on a graph
Rapid solution
[src]
Or(And(-oo < x, x < 2), And(7 < x, x < oo))
$$\left(-\infty < x \wedge x < 2\right) \vee \left(7 < x \wedge x < \infty\right)$$
((-oo < x)∧(x < 2))∨((7 < x)∧(x < oo))
Rapid solution 2
[src]
(-oo, 2) U (7, oo)
$$x\ in\ \left(-\infty, 2\right) \cup \left(7, \infty\right)$$
x in Union(Interval.open(-oo, 2), Interval.open(7, oo))