Mister Exam
(x-3)*(x-1)*lg(cos^2(pi*x)+1)/lg2<=-1 inequation
The solution
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/ 2 \
(x - 3)*(x - 1)*log\cos (pi*x) + 1/
----------------------------------- <= -1
log(2)
$$\frac{\left(x - 3\right) \left(x - 1\right) \log{\left(\cos^{2}{\left(\pi x \right)} + 1 \right)}}{\log{\left(2 \right)}} \leq -1$$
(((x - 3)*(x - 1))*log(cos(pi*x)^2 + 1))/log(2) <= -1
Detail solution
Given the inequality:
$$\frac{\left(x - 3\right) \left(x - 1\right) \log{\left(\cos^{2}{\left(\pi x \right)} + 1 \right)}}{\log{\left(2 \right)}} \leq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 3\right) \left(x - 1\right) \log{\left(\cos^{2}{\left(\pi x \right)} + 1 \right)}}{\log{\left(2 \right)}} = -1$$
Solve:
$$x_{1} = 1.99999977241718$$
$$x_{2} = 2.00000011425039$$
$$x_{3} = 2$$
$$x_{1} = 1.99999977241718$$
$$x_{2} = 2.00000011425039$$
$$x_{3} = 2$$
This roots
$$x_{1} = 1.99999977241718$$
$$x_{3} = 2$$
$$x_{2} = 2.00000011425039$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.99999977241718$$
=
$$1.89999977241718$$
substitute to the expression
$$\frac{\left(x - 3\right) \left(x - 1\right) \log{\left(\cos^{2}{\left(\pi x \right)} + 1 \right)}}{\log{\left(2 \right)}} \leq -1$$
$$\frac{\left(-3 + 1.89999977241718\right) \left(-1 + 1.89999977241718\right) \log{\left(\cos^{2}{\left(1.89999977241718 \pi \right)} + 1 \right)}}{\log{\left(2 \right)}} \leq -1$$
but
Then
$$x \leq 1.99999977241718$$
no execute
one of the solutions of our inequality is:
$$x \geq 1.99999977241718 \wedge x \leq 2$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq 1.99999977241718 \wedge x \leq 2$$
$$x \geq 2.00000011425039$$
$$\frac{\left(x - 3\right) \left(x - 1\right) \log{\left(\cos^{2}{\left(\pi x \right)} + 1 \right)}}{\log{\left(2 \right)}} \leq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 3\right) \left(x - 1\right) \log{\left(\cos^{2}{\left(\pi x \right)} + 1 \right)}}{\log{\left(2 \right)}} = -1$$
Solve:
$$x_{1} = 1.99999977241718$$
$$x_{2} = 2.00000011425039$$
$$x_{3} = 2$$
$$x_{1} = 1.99999977241718$$
$$x_{2} = 2.00000011425039$$
$$x_{3} = 2$$
This roots
$$x_{1} = 1.99999977241718$$
$$x_{3} = 2$$
$$x_{2} = 2.00000011425039$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.99999977241718$$
=
$$1.89999977241718$$
substitute to the expression
$$\frac{\left(x - 3\right) \left(x - 1\right) \log{\left(\cos^{2}{\left(\pi x \right)} + 1 \right)}}{\log{\left(2 \right)}} \leq -1$$
$$\frac{\left(-3 + 1.89999977241718\right) \left(-1 + 1.89999977241718\right) \log{\left(\cos^{2}{\left(1.89999977241718 \pi \right)} + 1 \right)}}{\log{\left(2 \right)}} \leq -1$$
/ 2 \
-0.989999954483384*log\1 + cos (1.89999977241718*pi)/
----------------------------------------------------- <= -1
log(2)
but
/ 2 \
-0.989999954483384*log\1 + cos (1.89999977241718*pi)/
----------------------------------------------------- >= -1
log(2)
Then
$$x \leq 1.99999977241718$$
no execute
one of the solutions of our inequality is:
$$x \geq 1.99999977241718 \wedge x \leq 2$$
_____ _____
/ \ /
-------•-------•-------•-------
x1 x3 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq 1.99999977241718 \wedge x \leq 2$$
$$x \geq 2.00000011425039$$
Solving inequality on a graph