Mister Exam
4^x+1-21*2^x+5>0 inequation
The solution
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x x 4 + 1 - 21*2 + 5 > 0
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 > 0$$
-21*2^x + 4^x + 1 + 5 > 0
Detail solution
Given the inequality:
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 = 0$$
Solve:
Given the equation:
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 = 0$$
or
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 = 0$$
Do replacement
$$v = 2^{x}$$
we get
$$v^{2} - 21 v + 6 = 0$$
or
$$v^{2} - 21 v + 6 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -21$$
$$c = 6$$
, then
Because D > 0, then the equation has two roots.
or
$$v_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
$$v_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
do backward replacement
$$2^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(2 \right)}}$$
$$x_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
$$x_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
$$x_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
$$x_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
This roots
$$x_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
$$x_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{21}{2} - \frac{\sqrt{417}}{2}\right)$$
=
$$\frac{52}{5} - \frac{\sqrt{417}}{2}$$
substitute to the expression
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 > 0$$
$$\left(- 21 \cdot 2^{\frac{52}{5} - \frac{\sqrt{417}}{2}} + \left(1 + 4^{\frac{52}{5} - \frac{\sqrt{417}}{2}}\right)\right) + 5 > 0$$
Then
$$x < \frac{21}{2} - \frac{\sqrt{417}}{2}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{21}{2} - \frac{\sqrt{417}}{2} \wedge x < \frac{\sqrt{417}}{2} + \frac{21}{2}$$
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 = 0$$
Solve:
Given the equation:
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 = 0$$
or
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 = 0$$
Do replacement
$$v = 2^{x}$$
we get
$$v^{2} - 21 v + 6 = 0$$
or
$$v^{2} - 21 v + 6 = 0$$
This equation is of the form
a*v^2 + b*v + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -21$$
$$c = 6$$
, then
D = b^2 - 4 * a * c =
(-21)^2 - 4 * (1) * (6) = 417
Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
$$v_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
do backward replacement
$$2^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(2 \right)}}$$
$$x_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
$$x_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
$$x_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
$$x_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
This roots
$$x_{2} = \frac{21}{2} - \frac{\sqrt{417}}{2}$$
$$x_{1} = \frac{\sqrt{417}}{2} + \frac{21}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{21}{2} - \frac{\sqrt{417}}{2}\right)$$
=
$$\frac{52}{5} - \frac{\sqrt{417}}{2}$$
substitute to the expression
$$\left(- 21 \cdot 2^{x} + \left(4^{x} + 1\right)\right) + 5 > 0$$
$$\left(- 21 \cdot 2^{\frac{52}{5} - \frac{\sqrt{417}}{2}} + \left(1 + 4^{\frac{52}{5} - \frac{\sqrt{417}}{2}}\right)\right) + 5 > 0$$
_____ _____
52 \/ 417 52 \/ 417
-- - ------- -- - ------- > 0
5 2 5 2
6 + 4 - 21*2 Then
$$x < \frac{21}{2} - \frac{\sqrt{417}}{2}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{21}{2} - \frac{\sqrt{417}}{2} \wedge x < \frac{\sqrt{417}}{2} + \frac{21}{2}$$
_____
/ \
-------ο-------ο-------
x2 x1
Solving inequality on a graph
Rapid solution
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/ / / _____\ \ / _____\\ | | |21 \/ 417 | | |21 \/ 417 || | | log|-- + -------| | log|-- - -------|| | | \2 2 / | \2 2 /| Or|And|x < oo, ----------------- < x|, x < -----------------| \ \ log(2) / log(2) /
$$\left(x < \infty \wedge \frac{\log{\left(\frac{\sqrt{417}}{2} + \frac{21}{2} \right)}}{\log{\left(2 \right)}} < x\right) \vee x < \frac{\log{\left(\frac{21}{2} - \frac{\sqrt{417}}{2} \right)}}{\log{\left(2 \right)}}$$
(x < log(21/2 - sqrt(417)/2)/log(2))∨((x < oo)∧(log(21/2 + sqrt(417)/2)/log(2) < x))
Rapid solution 2
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/ _____\ / _____\
|21 \/ 417 | |21 \/ 417 |
log|-- - -------| log|-- + -------|
\2 2 / \2 2 /
(-oo, -----------------) U (-----------------, oo)
log(2) log(2)
$$x\ in\ \left(-\infty, \frac{\log{\left(\frac{21}{2} - \frac{\sqrt{417}}{2} \right)}}{\log{\left(2 \right)}}\right) \cup \left(\frac{\log{\left(\frac{\sqrt{417}}{2} + \frac{21}{2} \right)}}{\log{\left(2 \right)}}, \infty\right)$$
x in Union(Interval.open(-oo, log(21/2 - sqrt(417)/2)/log(2)), Interval.open(log(sqrt(417)/2 + 21/2)/log(2), oo))