Mister Exam
10^(1-x)>0.0014 inequation
The solution
Detail solution
Given the inequality:
$$10^{1 - x} > 0.0014$$
To solve this inequality, we must first solve the corresponding equation:
$$10^{1 - x} = 0.0014$$
Solve:
Given the equation:
$$10^{1 - x} = 0.0014$$
or
$$10^{1 - x} - 0.0014 = 0$$
or
$$10 \cdot 10^{- x} = 0.0014$$
or
$$\left(\frac{1}{10}\right)^{x} = 0.00014$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{10}\right)^{x}$$
we get
$$v - 0.00014 = 0$$
or
$$v - 0.00014 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 0.00014$$
do backward replacement
$$\left(\frac{1}{10}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(10 \right)}}$$
$$x_{1} = 0.00014$$
$$x_{1} = 0.00014$$
This roots
$$x_{1} = 0.00014$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.00014$$
=
$$-0.09986$$
substitute to the expression
$$10^{1 - x} > 0.0014$$
$$10^{1 - -0.09986} > 0.0014$$
the solution of our inequality is:
$$x < 0.00014$$
$$10^{1 - x} > 0.0014$$
To solve this inequality, we must first solve the corresponding equation:
$$10^{1 - x} = 0.0014$$
Solve:
Given the equation:
$$10^{1 - x} = 0.0014$$
or
$$10^{1 - x} - 0.0014 = 0$$
or
$$10 \cdot 10^{- x} = 0.0014$$
or
$$\left(\frac{1}{10}\right)^{x} = 0.00014$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{10}\right)^{x}$$
we get
$$v - 0.00014 = 0$$
or
$$v - 0.00014 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 0.00014$$
do backward replacement
$$\left(\frac{1}{10}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(10 \right)}}$$
$$x_{1} = 0.00014$$
$$x_{1} = 0.00014$$
This roots
$$x_{1} = 0.00014$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.00014$$
=
$$-0.09986$$
substitute to the expression
$$10^{1 - x} > 0.0014$$
$$10^{1 - -0.09986} > 0.0014$$
12.5851964759505 > 0.00140000000000000
the solution of our inequality is:
$$x < 0.00014$$
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x1
Solving inequality on a graph
Rapid solution
[src]
6.57128304236092
x < 1 + ----------------
log(10)
$$x < 1 + \frac{6.57128304236092}{\log{\left(10 \right)}}$$
x < 1 + 6.57128304236092/log(10)
Rapid solution 2
[src]
6.57128304236092
(-oo, 1 + ----------------)
log(10)
$$x\ in\ \left(-\infty, 1 + \frac{6.57128304236092}{\log{\left(10 \right)}}\right)$$
x in Interval.open(-oo, 1 + 6.57128304236092/log(10))