Mister Exam
16^x-9<1 inequation
The solution
Detail solution
Given the inequality:
$$16^{x} - 9 < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$16^{x} - 9 = 1$$
Solve:
Given the equation:
$$16^{x} - 9 = 1$$
or
$$\left(16^{x} - 9\right) - 1 = 0$$
or
$$16^{x} = 10$$
or
$$16^{x} = 10$$
- this is the simplest exponential equation
Do replacement
$$v = 16^{x}$$
we get
$$v - 10 = 0$$
or
$$v - 10 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 10$$
do backward replacement
$$16^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(16 \right)}}$$
$$x_{1} = 10$$
$$x_{1} = 10$$
This roots
$$x_{1} = 10$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 10$$
=
$$\frac{99}{10}$$
substitute to the expression
$$16^{x} - 9 < 1$$
$$-9 + 16^{\frac{99}{10}} < 1$$
but
Then
$$x < 10$$
no execute
the solution of our inequality is:
$$x > 10$$
$$16^{x} - 9 < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$16^{x} - 9 = 1$$
Solve:
Given the equation:
$$16^{x} - 9 = 1$$
or
$$\left(16^{x} - 9\right) - 1 = 0$$
or
$$16^{x} = 10$$
or
$$16^{x} = 10$$
- this is the simplest exponential equation
Do replacement
$$v = 16^{x}$$
we get
$$v - 10 = 0$$
or
$$v - 10 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 10$$
do backward replacement
$$16^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(16 \right)}}$$
$$x_{1} = 10$$
$$x_{1} = 10$$
This roots
$$x_{1} = 10$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 10$$
=
$$\frac{99}{10}$$
substitute to the expression
$$16^{x} - 9 < 1$$
$$-9 + 16^{\frac{99}{10}} < 1$$
3/5
-9 + 549755813888*2 < 1
but
3/5
-9 + 549755813888*2 > 1
Then
$$x < 10$$
no execute
the solution of our inequality is:
$$x > 10$$
_____
/
-------ο-------
x1
Solving inequality on a graph
Rapid solution
[src]
log(10)
x < -------
log(16)
$$x < \frac{\log{\left(10 \right)}}{\log{\left(16 \right)}}$$
x < log(10)/log(16)
Rapid solution 2
[src]
log(10)
(-oo, -------)
log(16)
$$x\ in\ \left(-\infty, \frac{\log{\left(10 \right)}}{\log{\left(16 \right)}}\right)$$
x in Interval.open(-oo, log(10)/log(16))