Mister Exam
log(0.2x)<=1 inequation
The solution
Detail solution
Given the inequality:
$$\log{\left(\frac{x}{5} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{x}{5} \right)} = 1$$
Solve:
Given the equation
$$\log{\left(\frac{x}{5} \right)} = 1$$
$$\log{\left(\frac{x}{5} \right)} = 1$$
This equation is of the form:
By definition log
then
$$\frac{x}{5} = e^{1^{-1}}$$
simplify
$$\frac{x}{5} = e$$
$$x = 5 e$$
$$x_{1} = 5 e$$
$$x_{1} = 5 e$$
This roots
$$x_{1} = 5 e$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5 e$$
=
$$- \frac{1}{10} + 5 e$$
substitute to the expression
$$\log{\left(\frac{x}{5} \right)} \leq 1$$
$$\log{\left(\frac{- \frac{1}{10} + 5 e}{5} \right)} \leq 1$$
the solution of our inequality is:
$$x \leq 5 e$$
$$\log{\left(\frac{x}{5} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{x}{5} \right)} = 1$$
Solve:
Given the equation
$$\log{\left(\frac{x}{5} \right)} = 1$$
$$\log{\left(\frac{x}{5} \right)} = 1$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$\frac{x}{5} = e^{1^{-1}}$$
simplify
$$\frac{x}{5} = e$$
$$x = 5 e$$
$$x_{1} = 5 e$$
$$x_{1} = 5 e$$
This roots
$$x_{1} = 5 e$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5 e$$
=
$$- \frac{1}{10} + 5 e$$
substitute to the expression
$$\log{\left(\frac{x}{5} \right)} \leq 1$$
$$\log{\left(\frac{- \frac{1}{10} + 5 e}{5} \right)} \leq 1$$
log(-1/50 + E) <= 1
the solution of our inequality is:
$$x \leq 5 e$$
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Solving inequality on a graph