Mister Exam
log(0,5(x-1))+log(0,5(x-2))>=-1 inequation
The solution
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/x - 1\ /x - 2\ log|-----| + log|-----| >= -1 \ 2 / \ 2 /
$$\log{\left(\frac{x - 2}{2} \right)} + \log{\left(\frac{x - 1}{2} \right)} \geq -1$$
log((x - 2)/2) + log((x - 1)/2) >= -1
Detail solution
Given the inequality:
$$\log{\left(\frac{x - 2}{2} \right)} + \log{\left(\frac{x - 1}{2} \right)} \geq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{x - 2}{2} \right)} + \log{\left(\frac{x - 1}{2} \right)} = -1$$
Solve:
$$x_{1} = \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
$$x_{1} = \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
This roots
$$x_{1} = \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{\sqrt{e + 16}}{2 \sqrt{e^{1}}} + \frac{3}{2}\right)$$
=
$$\frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{7}{5}$$
substitute to the expression
$$\log{\left(\frac{x - 2}{2} \right)} + \log{\left(\frac{x - 1}{2} \right)} \geq -1$$
$$\log{\left(\frac{-2 + \left(\frac{\sqrt{e + 16}}{2 \sqrt{e^{1}}} + \frac{7}{5}\right)}{2} \right)} + \log{\left(\frac{-1 + \left(\frac{\sqrt{e + 16}}{2 \sqrt{e^{1}}} + \frac{7}{5}\right)}{2} \right)} \geq -1$$
but
Then
$$x \leq \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
no execute
the solution of our inequality is:
$$x \geq \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
$$\log{\left(\frac{x - 2}{2} \right)} + \log{\left(\frac{x - 1}{2} \right)} \geq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{x - 2}{2} \right)} + \log{\left(\frac{x - 1}{2} \right)} = -1$$
Solve:
$$x_{1} = \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
$$x_{1} = \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
This roots
$$x_{1} = \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{\sqrt{e + 16}}{2 \sqrt{e^{1}}} + \frac{3}{2}\right)$$
=
$$\frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{7}{5}$$
substitute to the expression
$$\log{\left(\frac{x - 2}{2} \right)} + \log{\left(\frac{x - 1}{2} \right)} \geq -1$$
$$\log{\left(\frac{-2 + \left(\frac{\sqrt{e + 16}}{2 \sqrt{e^{1}}} + \frac{7}{5}\right)}{2} \right)} + \log{\left(\frac{-1 + \left(\frac{\sqrt{e + 16}}{2 \sqrt{e^{1}}} + \frac{7}{5}\right)}{2} \right)} \geq -1$$
/ ________ -1/2\ / ________ -1/2\
| 3 \/ 16 + E *e | |1 \/ 16 + E *e |
log|- -- + ----------------| + log|- + ----------------| >= -1
\ 10 4 / \5 4 /
but
/ ________ -1/2\ / ________ -1/2\
| 3 \/ 16 + E *e | |1 \/ 16 + E *e |
log|- -- + ----------------| + log|- + ----------------| < -1
\ 10 4 / \5 4 /
Then
$$x \leq \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
no execute
the solution of our inequality is:
$$x \geq \frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}$$
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x1
Solving inequality on a graph
Rapid solution 2
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________ -1/2 3 \/ 16 + E *e [- + ----------------, oo) 2 2
$$x\ in\ \left[\frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2}, \infty\right)$$
x in Interval(sqrt(E + 16)*exp(-1/2)/2 + 3/2, oo)
Rapid solution
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________ -1/2 3 \/ 16 + E *e - + ---------------- <= x 2 2
$$\frac{\sqrt{e + 16}}{2 e^{\frac{1}{2}}} + \frac{3}{2} \leq x$$
sqrt(E + 16)*exp(-1/2)/2 + 3/2 <= x