Mister Exam
(4)^(4x+3)>=3/4 inequation
The solution
Detail solution
Given the inequality:
$$4^{4 x + 3} \geq \frac{3}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$4^{4 x + 3} = \frac{3}{4}$$
Solve:
Given the equation:
$$4^{4 x + 3} = \frac{3}{4}$$
or
$$4^{4 x + 3} - \frac{3}{4} = 0$$
or
$$64 \cdot 256^{x} = \frac{3}{4}$$
or
$$256^{x} = \frac{3}{256}$$
- this is the simplest exponential equation
Do replacement
$$v = 256^{x}$$
we get
$$v - \frac{3}{256} = 0$$
or
$$v - \frac{3}{256} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{3}{256}$$
do backward replacement
$$256^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(256 \right)}}$$
$$x_{1} = \frac{3}{256}$$
$$x_{1} = \frac{3}{256}$$
This roots
$$x_{1} = \frac{3}{256}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3}{256}$$
=
$$- \frac{113}{1280}$$
substitute to the expression
$$4^{4 x + 3} \geq \frac{3}{4}$$
$$4^{\frac{\left(-113\right) 4}{1280} + 3} \geq \frac{3}{4}$$
the solution of our inequality is:
$$x \leq \frac{3}{256}$$
$$4^{4 x + 3} \geq \frac{3}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$4^{4 x + 3} = \frac{3}{4}$$
Solve:
Given the equation:
$$4^{4 x + 3} = \frac{3}{4}$$
or
$$4^{4 x + 3} - \frac{3}{4} = 0$$
or
$$64 \cdot 256^{x} = \frac{3}{4}$$
or
$$256^{x} = \frac{3}{256}$$
- this is the simplest exponential equation
Do replacement
$$v = 256^{x}$$
we get
$$v - \frac{3}{256} = 0$$
or
$$v - \frac{3}{256} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{3}{256}$$
do backward replacement
$$256^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(256 \right)}}$$
$$x_{1} = \frac{3}{256}$$
$$x_{1} = \frac{3}{256}$$
This roots
$$x_{1} = \frac{3}{256}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3}{256}$$
=
$$- \frac{113}{1280}$$
substitute to the expression
$$4^{4 x + 3} \geq \frac{3}{4}$$
$$4^{\frac{\left(-113\right) 4}{1280} + 3} \geq \frac{3}{4}$$
47
---
160 >= 3/4
32*2
the solution of our inequality is:
$$x \leq \frac{3}{256}$$
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x1
Solving inequality on a graph
Rapid solution 2
[src]
log(3)
[-1 + --------, oo)
4*log(4)
$$x\ in\ \left[-1 + \frac{\log{\left(3 \right)}}{4 \log{\left(4 \right)}}, \infty\right)$$
x in Interval(-1 + log(3)/(4*log(4)), oo)
Rapid solution
[src]
log(3)
-1 + -------- <= x
4*log(4)
$$-1 + \frac{\log{\left(3 \right)}}{4 \log{\left(4 \right)}} \leq x$$
-1 + log(3)/(4*log(4)) <= x