Mister Exam
2^(2-x^2)≥3 inequation
The solution
Detail solution
Given the inequality:
$$2^{2 - x^{2}} \geq 3$$
To solve this inequality, we must first solve the corresponding equation:
$$2^{2 - x^{2}} = 3$$
Solve:
$$x_{1} = - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
This roots
$$x_{1} = - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} + - \frac{1}{10}$$
=
$$- \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} - \frac{1}{10}$$
substitute to the expression
$$2^{2 - x^{2}} \geq 3$$
$$2^{2 - \left(- \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} - \frac{1}{10}\right)^{2}} \geq 3$$
but
Then
$$x \leq - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} \wedge x \leq \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$2^{2 - x^{2}} \geq 3$$
To solve this inequality, we must first solve the corresponding equation:
$$2^{2 - x^{2}} = 3$$
Solve:
$$x_{1} = - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{1} = - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
This roots
$$x_{1} = - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} + - \frac{1}{10}$$
=
$$- \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} - \frac{1}{10}$$
substitute to the expression
$$2^{2 - x^{2}} \geq 3$$
$$2^{2 - \left(- \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} - \frac{1}{10}\right)^{2}} \geq 3$$
2
/ __________\
| 1 \/ log(4/3) |
2 - |- -- - ------------| >= 3
| 10 ________ |
\ \/ log(2) /
2 but
2
/ __________\
| 1 \/ log(4/3) |
2 - |- -- - ------------| < 3
| 10 ________ |
\ \/ log(2) /
2 Then
$$x \leq - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}} \wedge x \leq \frac{\sqrt{\log{\left(\frac{4}{3} \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
_____
/ \
-------•-------•-------
x1 x2
Solving inequality on a graph
Rapid solution 2
[src]
____________ ____________
/ log(3) / log(3)
[- / 2 - ------ , / 2 - ------ ]
\/ log(2) \/ log(2)
$$x\ in\ \left[- \sqrt{- \frac{\log{\left(3 \right)}}{\log{\left(2 \right)}} + 2}, \sqrt{- \frac{\log{\left(3 \right)}}{\log{\left(2 \right)}} + 2}\right]$$
x in Interval(-sqrt(-log(3)/log(2) + 2), sqrt(-log(3)/log(2) + 2))
Rapid solution
[src]
/ ____________ ____________ \ | / log(3) / log(3) | And|x <= / 2 - ------ , - / 2 - ------ <= x| \ \/ log(2) \/ log(2) /
$$x \leq \sqrt{- \frac{\log{\left(3 \right)}}{\log{\left(2 \right)}} + 2} \wedge - \sqrt{- \frac{\log{\left(3 \right)}}{\log{\left(2 \right)}} + 2} \leq x$$
(x <= sqrt(2 - log(3)/log(2)))∧(-sqrt(2 - log(3)/log(2)) <= x)