Mister Exam
log((4-x)/(2x+1))<=log((4-x)/8)+log((4-x)/(x^2)) inequation
The solution
You have entered
[src]
/ 4 - x \ /4 - x\ /4 - x\
log|-------| <= log|-----| + log|-----|
\2*x + 1/ \ 8 / | 2 |
\ x /
$$\log{\left(\frac{- x + 4}{2 x + 1} \right)} \leq \log{\left(\frac{- x + 4}{8} \right)} + \log{\left(\frac{- x + 4}{x^{2}} \right)}$$
log((4 - x)/(2*x + 1)) <= log(4 - x/8) + log((4 - x)/(x^2))
Detail solution
Given the inequality:
$$\log{\left(\frac{- x + 4}{2 x + 1} \right)} \leq \log{\left(\frac{- x + 4}{8} \right)} + \log{\left(\frac{- x + 4}{x^{2}} \right)}$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{- x + 4}{2 x + 1} \right)} = \log{\left(\frac{- x + 4}{8} \right)} + \log{\left(\frac{- x + 4}{x^{2}} \right)}$$
Solve:
$$x_{1} = 1.07284161474005$$
$$x_{1} = 1.07284161474005$$
This roots
$$x_{1} = 1.07284161474005$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.07284161474005$$
=
$$0.972841614740048$$
substitute to the expression
$$\log{\left(\frac{- x + 4}{2 x + 1} \right)} \leq \log{\left(\frac{- x + 4}{8} \right)} + \log{\left(\frac{- x + 4}{x^{2}} \right)}$$
$$\log{\left(\frac{\left(-1\right) 0.972841614740048 + 4}{1 + 2 \cdot 0.972841614740048} \right)} \leq \log{\left(\frac{\left(-1\right) 0.972841614740048 + 4}{8} \right)} + \log{\left(\frac{\left(-1\right) 0.972841614740048 + 4}{0.972841614740048^{2}} \right)}$$
the solution of our inequality is:
$$x \leq 1.07284161474005$$
$$\log{\left(\frac{- x + 4}{2 x + 1} \right)} \leq \log{\left(\frac{- x + 4}{8} \right)} + \log{\left(\frac{- x + 4}{x^{2}} \right)}$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{- x + 4}{2 x + 1} \right)} = \log{\left(\frac{- x + 4}{8} \right)} + \log{\left(\frac{- x + 4}{x^{2}} \right)}$$
Solve:
$$x_{1} = 1.07284161474005$$
$$x_{1} = 1.07284161474005$$
This roots
$$x_{1} = 1.07284161474005$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.07284161474005$$
=
$$0.972841614740048$$
substitute to the expression
$$\log{\left(\frac{- x + 4}{2 x + 1} \right)} \leq \log{\left(\frac{- x + 4}{8} \right)} + \log{\left(\frac{- x + 4}{x^{2}} \right)}$$
$$\log{\left(\frac{\left(-1\right) 0.972841614740048 + 4}{1 + 2 \cdot 0.972841614740048} \right)} \leq \log{\left(\frac{\left(-1\right) 0.972841614740048 + 4}{8} \right)} + \log{\left(\frac{\left(-1\right) 0.972841614740048 + 4}{0.972841614740048^{2}} \right)}$$
0.0272835662154318 <= 0.190875144750097
the solution of our inequality is:
$$x \leq 1.07284161474005$$
_____
\
-------•-------
x_1
Solving inequality on a graph
Rapid solution
[src]
/ _____ _____ \ | 7 \/ 209 7 \/ 209 | And|x > -- - -------, x < -- + -------, x != 0| \ 20 20 20 20 /
$$x > - \frac{\sqrt{209}}{20} + \frac{7}{20} \wedge x < \frac{7}{20} + \frac{\sqrt{209}}{20} \wedge x \neq 0$$
(Ne(x, 0))∧(x > 7/20 - sqrt(209)/20)∧(x < 7/20 + sqrt(209)/20)
Rapid solution 2
[src]
_____ _____ 7 \/ 209 7 \/ 209 (-- - -------, 0) U (0, -- + -------) 20 20 20 20
$$x\ in\ \left(- \frac{\sqrt{209}}{20} + \frac{7}{20}, 0\right) \cup \left(0, \frac{7}{20} + \frac{\sqrt{209}}{20}\right)$$
x in Union(Interval.open(0, 7/20 + sqrt(209)/20), Interval.open(7/20 - sqrt(209)/20, 0))