Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- (x-5)*(sqrt(9-x))>0
- 3^(log(x-1/x+2))>1/9
- 4^(x^2-x-6)-1>0
- 2^(x^2)*13^(x-1)>=2
- Limit of the function:
-
(1/5)^x
- Integral of d{x}:
- (1/5)^x
-
Identical expressions
- (one / five)^x< four
- (1 divide by 5) to the power of x less than 4
- (one divide by five) to the power of x less than four
- (1/5)x<4
- 1/5x<4
- 1/5^x<4
- (1 divide by 5)^x<4
(1/5)^x<4 inequation
The solution
Detail solution
Given the inequality:
$$\left(\frac{1}{5}\right)^{x} < 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{1}{5}\right)^{x} = 4$$
Solve:
Given the equation:
$$\left(\frac{1}{5}\right)^{x} = 4$$
or
$$-4 + \left(\frac{1}{5}\right)^{x} = 0$$
or
$$\left(\frac{1}{5}\right)^{x} = 4$$
or
$$\left(\frac{1}{5}\right)^{x} = 4$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{5}\right)^{x}$$
we get
$$v - 4 = 0$$
or
$$v - 4 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 4$$
do backward replacement
$$\left(\frac{1}{5}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}$$
$$x_{1} = 4$$
$$x_{1} = 4$$
This roots
$$x_{1} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 4$$
=
$$\frac{39}{10}$$
substitute to the expression
$$\left(\frac{1}{5}\right)^{x} < 4$$
$$\left(\frac{1}{5}\right)^{\frac{39}{10}} < 4$$
the solution of our inequality is:
$$x < 4$$
$$\left(\frac{1}{5}\right)^{x} < 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{1}{5}\right)^{x} = 4$$
Solve:
Given the equation:
$$\left(\frac{1}{5}\right)^{x} = 4$$
or
$$-4 + \left(\frac{1}{5}\right)^{x} = 0$$
or
$$\left(\frac{1}{5}\right)^{x} = 4$$
or
$$\left(\frac{1}{5}\right)^{x} = 4$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{5}\right)^{x}$$
we get
$$v - 4 = 0$$
or
$$v - 4 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 4$$
do backward replacement
$$\left(\frac{1}{5}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}$$
$$x_{1} = 4$$
$$x_{1} = 4$$
This roots
$$x_{1} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 4$$
=
$$\frac{39}{10}$$
substitute to the expression
$$\left(\frac{1}{5}\right)^{x} < 4$$
$$\left(\frac{1}{5}\right)^{\frac{39}{10}} < 4$$
10___
\/ 5
----- < 4
625
the solution of our inequality is:
$$x < 4$$
_____
\
-------ο-------
x1
Solving inequality on a graph
Rapid solution
[src]
-log(4) -------- < x log(5)
$$- \frac{\log{\left(4 \right)}}{\log{\left(5 \right)}} < x$$
-log(4)/log(5) < x
Rapid solution 2
[src]
-log(4) (--------, oo) log(5)
$$x\ in\ \left(- \frac{\log{\left(4 \right)}}{\log{\left(5 \right)}}, \infty\right)$$
x in Interval.open(-log(4)/log(5), oo)