Mister Exam
(1/2)^(2x+1)<4 inequation
The solution
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-1 - 2*x 2 < 4
$$\left(\frac{1}{2}\right)^{2 x + 1} < 4$$
(1/2)^(2*x + 1) < 4
Detail solution
Given the inequality:
$$\left(\frac{1}{2}\right)^{2 x + 1} < 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{1}{2}\right)^{2 x + 1} = 4$$
Solve:
Given the equation:
$$\left(\frac{1}{2}\right)^{2 x + 1} = 4$$
or
$$\left(\frac{1}{2}\right)^{2 x + 1} - 4 = 0$$
or
$$\frac{4^{- x}}{2} = 4$$
or
$$\left(\frac{1}{4}\right)^{x} = 8$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{4}\right)^{x}$$
we get
$$v - 8 = 0$$
or
$$v - 8 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 8$$
do backward replacement
$$\left(\frac{1}{4}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(4 \right)}}$$
$$x_{1} = 8$$
$$x_{1} = 8$$
This roots
$$x_{1} = 8$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 8$$
=
$$\frac{79}{10}$$
substitute to the expression
$$\left(\frac{1}{2}\right)^{2 x + 1} < 4$$
$$\left(\frac{1}{2}\right)^{1 + \frac{2 \cdot 79}{10}} < 4$$
the solution of our inequality is:
$$x < 8$$
$$\left(\frac{1}{2}\right)^{2 x + 1} < 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{1}{2}\right)^{2 x + 1} = 4$$
Solve:
Given the equation:
$$\left(\frac{1}{2}\right)^{2 x + 1} = 4$$
or
$$\left(\frac{1}{2}\right)^{2 x + 1} - 4 = 0$$
or
$$\frac{4^{- x}}{2} = 4$$
or
$$\left(\frac{1}{4}\right)^{x} = 8$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{4}\right)^{x}$$
we get
$$v - 8 = 0$$
or
$$v - 8 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 8$$
do backward replacement
$$\left(\frac{1}{4}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(4 \right)}}$$
$$x_{1} = 8$$
$$x_{1} = 8$$
This roots
$$x_{1} = 8$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 8$$
=
$$\frac{79}{10}$$
substitute to the expression
$$\left(\frac{1}{2}\right)^{2 x + 1} < 4$$
$$\left(\frac{1}{2}\right)^{1 + \frac{2 \cdot 79}{10}} < 4$$
5 ___
\/ 2
------ < 4
131072
the solution of our inequality is:
$$x < 8$$
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x1
Solving inequality on a graph
Rapid solution
[src]
1 log(4) - - - -------- < x 2 2*log(2)
$$- \frac{\log{\left(4 \right)}}{2 \log{\left(2 \right)}} - \frac{1}{2} < x$$
-log(4)/(2*log(2)) - 1/2 < x
Rapid solution 2
[src]
1 log(4) (- - - --------, oo) 2 2*log(2)
$$x\ in\ \left(- \frac{\log{\left(4 \right)}}{2 \log{\left(2 \right)}} - \frac{1}{2}, \infty\right)$$
x in Interval.open(-log(4)/(2*log(2)) - 1/2, oo)