Mister Exam
(5/(sqrtx-1)-5)>=-1 inequation
The solution
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5 --------- - 5 >= -1 ___ \/ x - 1
$$-5 + \frac{5}{\sqrt{x} - 1} \geq -1$$
-5 + 5/(sqrt(x) - 1) >= -1
Detail solution
Given the inequality:
$$-5 + \frac{5}{\sqrt{x} - 1} \geq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$-5 + \frac{5}{\sqrt{x} - 1} = -1$$
Solve:
$$x_{1} = \frac{81}{16}$$
$$x_{1} = \frac{81}{16}$$
This roots
$$x_{1} = \frac{81}{16}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{81}{16}$$
=
$$\frac{397}{80}$$
substitute to the expression
$$-5 + \frac{5}{\sqrt{x} - 1} \geq -1$$
$$-5 + \frac{5}{-1 + \sqrt{\frac{397}{80}}} \geq -1$$
the solution of our inequality is:
$$x \leq \frac{81}{16}$$
$$-5 + \frac{5}{\sqrt{x} - 1} \geq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$-5 + \frac{5}{\sqrt{x} - 1} = -1$$
Solve:
$$x_{1} = \frac{81}{16}$$
$$x_{1} = \frac{81}{16}$$
This roots
$$x_{1} = \frac{81}{16}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{81}{16}$$
=
$$\frac{397}{80}$$
substitute to the expression
$$-5 + \frac{5}{\sqrt{x} - 1} \geq -1$$
$$-5 + \frac{5}{-1 + \sqrt{\frac{397}{80}}} \geq -1$$
5
-5 + -------------
______
\/ 1985 >= -1
-1 + --------
20
the solution of our inequality is:
$$x \leq \frac{81}{16}$$
_____
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x1
Solving inequality on a graph
Rapid solution
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/ 81 \ And|x <= --, 1 < x| \ 16 /
$$x \leq \frac{81}{16} \wedge 1 < x$$
(x <= 81/16)∧(1 < x)
Rapid solution 2
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81
(1, --]
16
$$x\ in\ \left(1, \frac{81}{16}\right]$$
x in Interval.Lopen(1, 81/16)