Mister Exam
sqrt(3^(x-54))-7*sqrt(3^(x-58))<162 inequation
The solution
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_________ _________ / x - 54 / x - 58 \/ 3 - 7*\/ 3 < 162
$$\sqrt{3^{x - 54}} - 7 \sqrt{3^{x - 58}} < 162$$
sqrt(3^(x - 1*54)) - 7*sqrt(3^(x - 1*58)) < 162
Detail solution
Given the inequality:
$$\sqrt{3^{x - 54}} - 7 \sqrt{3^{x - 58}} < 162$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3^{x - 54}} - 7 \sqrt{3^{x - 58}} = 162$$
Solve:
$$x_{1} = 66$$
$$x_{1} = 66$$
This roots
$$x_{1} = 66$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 66$$
=
$$\frac{659}{10}$$
substitute to the expression
$$\sqrt{3^{x - 54}} - 7 \sqrt{3^{x - 58}} < 162$$
$$- 7 \sqrt{3^{\frac{659}{10} - 58}} + \sqrt{3^{\frac{659}{10} - 54}} < 162$$
the solution of our inequality is:
$$x < 66$$
$$\sqrt{3^{x - 54}} - 7 \sqrt{3^{x - 58}} < 162$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3^{x - 54}} - 7 \sqrt{3^{x - 58}} = 162$$
Solve:
$$x_{1} = 66$$
$$x_{1} = 66$$
This roots
$$x_{1} = 66$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 66$$
=
$$\frac{659}{10}$$
substitute to the expression
$$\sqrt{3^{x - 54}} - 7 \sqrt{3^{x - 58}} < 162$$
$$- 7 \sqrt{3^{\frac{659}{10} - 58}} + \sqrt{3^{\frac{659}{10} - 54}} < 162$$
19
--
20 ___ 9/20 < 162
- 189*3 + 243*\/ 3 *3
the solution of our inequality is:
$$x < 66$$
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x_1
Rapid solution
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/ 2*log(5559060566555523)\ And|-oo < x, x < -----------------------| \ log(3) /
$$-\infty < x \wedge x < \frac{2 \log{\left(5559060566555523 \right)}}{\log{\left(3 \right)}}$$
(-oo < x)∧(x < 2*log(5559060566555523)/log(3))
Rapid solution 2
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2*log(5559060566555523)
(-oo, -----------------------)
log(3)
$$x\ in\ \left(-\infty, \frac{2 \log{\left(5559060566555523 \right)}}{\log{\left(3 \right)}}\right)$$
x in Interval.open(-oo, 2*log(5559060566555523)/log(3))