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Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- log((x-8)^10)/log(x-1)>=10
- (2-x)/(2x-8)>0
- (2-x)(x+4)²x³/x+1<0
- 5x-3x^2-11<=0
-
Identical expressions
- six * five ^x- eleven / twenty-five ^(x- zero , five)- six * five ^(x)+ one >= zero , twenty-five
- 6 multiply by 5 to the power of x minus 11 divide by 25 to the power of (x minus 0,5) minus 6 multiply by 5 to the power of (x) plus 1 greater than or equal to 0,25
- six multiply by five to the power of x minus eleven divide by twenty minus five to the power of (x minus zero , five) minus six multiply by five to the power of (x) plus one greater than or equal to zero , twenty minus five
- 6*5x-11/25(x-0,5)-6*5(x)+1>=0,25
- 6*5x-11/25x-0,5-6*5x+1>=0,25
- 65^x-11/25^(x-0,5)-65^(x)+1>=0,25
- 65x-11/25(x-0,5)-65(x)+1>=0,25
- 65x-11/25x-0,5-65x+1>=0,25
- 65^x-11/25^x-0,5-65^x+1>=0,25
- 6*5^x-11/25^(x-0,5)-6*5^(x)+1>=O,25
- 6*5^x-11 divide by 25^(x-0,5)-6*5^(x)+1>=0,25
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Similar expressions
- 6*5^x+11/25^(x-0,5)-6*5^(x)+1>=0,25
- 6*5^x-11/25^(x-0,5)-6*5^(x)-1>=0,25
- 6*5^x-11/25^(x+0,5)-6*5^(x)+1>=0,25
- 6*5^x-11/25^(x-0,5)+6*5^(x)+1>=0,25
6*5^x-11/25^(x-0,5)-6*5^(x)+1>=0,25 inequation
The solution
You have entered
[src]
x - 1/2
x /11\ x
6*5 - |--| - 6*5 + 1 >= 1/4
\25/
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 \geq \frac{1}{4}$$
-(11/25)^(x - 1*1/2) - 6*5^x + 6*5^x + 1 >= 1/4
Detail solution
Given the inequality:
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 \geq \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 = \frac{1}{4}$$
Solve:
Given the equation:
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 = \frac{1}{4}$$
or
$$\left(- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1\right) - \frac{1}{4} = 0$$
or
$$- \frac{5 \sqrt{11} \left(\frac{11}{25}\right)^{x}}{11} = - \frac{3}{4}$$
or
$$\left(\frac{11}{25}\right)^{x} = \frac{3 \sqrt{11}}{20}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{11}{25}\right)^{x}$$
we get
$$v - \frac{3 \sqrt{11}}{20} = 0$$
or
$$v - \frac{3 \sqrt{11}}{20} = 0$$
Expand brackets in the left part
Divide both parts of the equation by (v - 3*sqrt(11)/20)/v
do backward replacement
$$\left(\frac{11}{25}\right)^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(\frac{11}{25} \right)}}$$
$$x_{1} = \frac{3 \sqrt{11}}{20}$$
$$x_{1} = \frac{3 \sqrt{11}}{20}$$
This roots
$$x_{1} = \frac{3 \sqrt{11}}{20}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3 \sqrt{11}}{20}$$
=
$$- \frac{1}{10} + \frac{3 \sqrt{11}}{20}$$
substitute to the expression
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 \geq \frac{1}{4}$$
$$- 6 \cdot 5^{- \frac{1}{10} + \frac{3 \sqrt{11}}{20}} - \left(\frac{11}{25}\right)^{\left(-1\right) \frac{1}{2} - \left(- \frac{3 \sqrt{11}}{20} + \frac{1}{10}\right)} + 1 + 6 \cdot 5^{- \frac{1}{10} + \frac{3 \sqrt{11}}{20}} \geq \frac{1}{4}$$
but
Then
$$x \leq \frac{3 \sqrt{11}}{20}$$
no execute
the solution of our inequality is:
$$x \geq \frac{3 \sqrt{11}}{20}$$
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 \geq \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 = \frac{1}{4}$$
Solve:
Given the equation:
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 = \frac{1}{4}$$
or
$$\left(- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1\right) - \frac{1}{4} = 0$$
or
$$- \frac{5 \sqrt{11} \left(\frac{11}{25}\right)^{x}}{11} = - \frac{3}{4}$$
or
$$\left(\frac{11}{25}\right)^{x} = \frac{3 \sqrt{11}}{20}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{11}{25}\right)^{x}$$
we get
$$v - \frac{3 \sqrt{11}}{20} = 0$$
or
$$v - \frac{3 \sqrt{11}}{20} = 0$$
Expand brackets in the left part
v - 3*sqrt11/20 = 0
Divide both parts of the equation by (v - 3*sqrt(11)/20)/v
v = 0 / ((v - 3*sqrt(11)/20)/v)
do backward replacement
$$\left(\frac{11}{25}\right)^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(\frac{11}{25} \right)}}$$
$$x_{1} = \frac{3 \sqrt{11}}{20}$$
$$x_{1} = \frac{3 \sqrt{11}}{20}$$
This roots
$$x_{1} = \frac{3 \sqrt{11}}{20}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3 \sqrt{11}}{20}$$
=
$$- \frac{1}{10} + \frac{3 \sqrt{11}}{20}$$
substitute to the expression
$$- \left(\frac{11}{25}\right)^{x - \frac{1}{2}} - 6 \cdot 5^{x} + 6 \cdot 5^{x} + 1 \geq \frac{1}{4}$$
$$- 6 \cdot 5^{- \frac{1}{10} + \frac{3 \sqrt{11}}{20}} - \left(\frac{11}{25}\right)^{\left(-1\right) \frac{1}{2} - \left(- \frac{3 \sqrt{11}}{20} + \frac{1}{10}\right)} + 1 + 6 \cdot 5^{- \frac{1}{10} + \frac{3 \sqrt{11}}{20}} \geq \frac{1}{4}$$
____
3 3*\/ 11
- - + --------
5 20 >= 1/4
/11\
1 - |--|
\25/ but
____
3 3*\/ 11
- - + --------
5 20 < 1/4
/11\
1 - |--|
\25/ Then
$$x \leq \frac{3 \sqrt{11}}{20}$$
no execute
the solution of our inequality is:
$$x \geq \frac{3 \sqrt{11}}{20}$$
_____
/
-------•-------
x_1
Solving inequality on a graph
Rapid solution
[src]
1 log(3/4)
- + -------- <= x
2 /11\
log|--|
\25/
$$\frac{\log{\left(\frac{3}{4} \right)}}{\log{\left(\frac{11}{25} \right)}} + \frac{1}{2} \leq x$$
log(3/4)/log(11/25) + 1/2 <= x
Rapid solution 2
[src]
1 log(3/4)
[- + --------, oo)
2 /11\
log|--|
\25/
$$x\ in\ \left[\frac{\log{\left(\frac{3}{4} \right)}}{\log{\left(\frac{11}{25} \right)}} + \frac{1}{2}, \infty\right)$$
x in Interval(log(3/4)/log(11/25) + 1/2, oo)
The graph