Mister Exam
|log5/log0.2|<|logx/log0.2| inequation
The solution
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| log(5) | | log(x) | |--------| < |--------| |log(0.2)| |log(0.2)|
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| < \left|{\frac{\log{\left(x \right)}}{\log{\left(0.2 \right)}}}\right|$$
Abs(log(5)/log(0.2)) < Abs(log(x)/log(0.2))
Detail solution
Given the inequality:
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| < \left|{\frac{\log{\left(x \right)}}{\log{\left(0.2 \right)}}}\right|$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| = \left|{\frac{\log{\left(x \right)}}{\log{\left(0.2 \right)}}}\right|$$
Solve:
$$x_{1} = 5$$
$$x_{1} = 5$$
This roots
$$x_{1} = 5$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$4.9$$
substitute to the expression
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| < \left|{\frac{\log{\left(x \right)}}{\log{\left(0.2 \right)}}}\right|$$
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| < \left|{\frac{\log{\left(4.9 \right)}}{\log{\left(0.2 \right)}}}\right|$$
but
Then
$$x < 5$$
no execute
the solution of our inequality is:
$$x > 5$$
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| < \left|{\frac{\log{\left(x \right)}}{\log{\left(0.2 \right)}}}\right|$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| = \left|{\frac{\log{\left(x \right)}}{\log{\left(0.2 \right)}}}\right|$$
Solve:
$$x_{1} = 5$$
$$x_{1} = 5$$
This roots
$$x_{1} = 5$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$4.9$$
substitute to the expression
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| < \left|{\frac{\log{\left(x \right)}}{\log{\left(0.2 \right)}}}\right|$$
$$\left|{\frac{\log{\left(5 \right)}}{\log{\left(0.2 \right)}}}\right| < \left|{\frac{\log{\left(4.9 \right)}}{\log{\left(0.2 \right)}}}\right|$$
0.621334934559612*log(5) < 0.987447352170942
but
0.621334934559612*log(5) > 0.987447352170942
Then
$$x < 5$$
no execute
the solution of our inequality is:
$$x > 5$$
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Solving inequality on a graph
Rapid solution
[src]
Or(And(0 <= x, x < 0.2), 5.0 < x)
$$\left(0 \leq x \wedge x < 0.2\right) \vee 5.0 < x$$
(5.0 < x)∨((0 <= x)∧(x < 0.2))
Rapid solution 2
[src]
[0, 0.2) U (5.0, oo)
$$x\ in\ \left[0, 0.2\right) \cup \left(5.0, \infty\right)$$
x in Union(Interval.Ropen(0, 0.200000000000000), Interval.open(5.00000000000000, oo))