Mister Exam
log(x*(6*x-5/(7)))>1 inequation
The solution
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[src]
log(x*(6*x - 0.714285714285714)) > 1
$$\log{\left(x \left(6 x - 0.714285714285714\right) \right)} > 1$$
log(x*(6*x - 0.714285714285714)) > 1
Detail solution
Given the inequality:
$$\log{\left(x \left(6 x - 0.714285714285714\right) \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(x \left(6 x - 0.714285714285714\right) \right)} = 1$$
Solve:
$$x_{1} = -0.616190668129339$$
$$x_{2} = 0.735238287176958$$
$$x_{1} = -0.616190668129339$$
$$x_{2} = 0.735238287176958$$
This roots
$$x_{1} = -0.616190668129339$$
$$x_{2} = 0.735238287176958$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-0.616190668129339 + - \frac{1}{10}$$
=
$$-0.716190668129339$$
substitute to the expression
$$\log{\left(x \left(6 x - 0.714285714285714\right) \right)} > 1$$
$$\log{\left(- 0.716190668129339 \left(\left(-0.716190668129339\right) 6 - 0.714285714285714\right) \right)} > 1$$
one of the solutions of our inequality is:
$$x < -0.616190668129339$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -0.616190668129339$$
$$x > 0.735238287176958$$
$$\log{\left(x \left(6 x - 0.714285714285714\right) \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(x \left(6 x - 0.714285714285714\right) \right)} = 1$$
Solve:
$$x_{1} = -0.616190668129339$$
$$x_{2} = 0.735238287176958$$
$$x_{1} = -0.616190668129339$$
$$x_{2} = 0.735238287176958$$
This roots
$$x_{1} = -0.616190668129339$$
$$x_{2} = 0.735238287176958$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-0.616190668129339 + - \frac{1}{10}$$
=
$$-0.716190668129339$$
substitute to the expression
$$\log{\left(x \left(6 x - 0.714285714285714\right) \right)} > 1$$
$$\log{\left(- 0.716190668129339 \left(\left(-0.716190668129339\right) 6 - 0.714285714285714\right) \right)} > 1$$
1.27791239704800 > 1
one of the solutions of our inequality is:
$$x < -0.616190668129339$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -0.616190668129339$$
$$x > 0.735238287176958$$
Solving inequality on a graph
Rapid solution
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/ ____________________________ ____________________________ \ Or\x < 0.0595238095238095 - 0.408248290463863*\/ 0.0212585034013605 + 1.0*E , 0.0595238095238095 + 0.408248290463863*\/ 0.0212585034013605 + 1.0*E < x/
$$x < 0.0595238095238095 - 0.408248290463863 \sqrt{0.0212585034013605 + 1.0 e} \vee 0.0595238095238095 + 0.408248290463863 \sqrt{0.0212585034013605 + 1.0 e} < x$$
(x < 0.0595238095238095 - 0.408248290463863*sqrt(0.0212585034013605 + 1.0*E))∨(0.0595238095238095 + 0.408248290463863*sqrt(0.0212585034013605 + 1.0*E) < x)
Rapid solution 2
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____________________________ ____________________________ (-oo, 0.0595238095238095 - 0.408248290463863*\/ 0.0212585034013605 + 1.0*E ) U (0.0595238095238095 + 0.408248290463863*\/ 0.0212585034013605 + 1.0*E , oo)
$$x\ in\ \left(-\infty, 0.0595238095238095 - 0.408248290463863 \sqrt{0.0212585034013605 + 1.0 e}\right) \cup \left(0.0595238095238095 + 0.408248290463863 \sqrt{0.0212585034013605 + 1.0 e}, \infty\right)$$
x in Union(Interval.open(-oo, 0.0595238095238095 - 0.408248290463863*sqrt(0.0212585034013605 + 1.0*E)), Interval.open(0.0595238095238095 + 0.408248290463863*sqrt(0.0212585034013605 + 1.0*E), oo))