Mister Exam
log((2x+4),((x^2)-x))>1 inequation
The solution
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/ 2 \ log\2*x + 4, x - x/ > 1
$$\log{\left(2 x + 4 \right)} > 1$$
log(2*x + 4, x^2 - x) > 1
Detail solution
Given the inequality:
$$\log{\left(2 x + 4 \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(2 x + 4 \right)} = 1$$
Solve:
$$x_{1} = -1$$
$$x_{2} = 4$$
$$x_{1} = -1$$
$$x_{2} = 4$$
This roots
$$x_{1} = -1$$
$$x_{2} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1 + - \frac{1}{10}$$
=
$$- \frac{11}{10}$$
substitute to the expression
$$\log{\left(2 x + 4 \right)} > 1$$
$$\log{\left(\frac{\left(-11\right) 2}{10} + 4 \right)} > 1$$
Then
$$x < -1$$
no execute
one of the solutions of our inequality is:
$$x > -1 \wedge x < 4$$
$$\log{\left(2 x + 4 \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(2 x + 4 \right)} = 1$$
Solve:
$$x_{1} = -1$$
$$x_{2} = 4$$
$$x_{1} = -1$$
$$x_{2} = 4$$
This roots
$$x_{1} = -1$$
$$x_{2} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1 + - \frac{1}{10}$$
=
$$- \frac{11}{10}$$
substitute to the expression
$$\log{\left(2 x + 4 \right)} > 1$$
$$\log{\left(\frac{\left(-11\right) 2}{10} + 4 \right)} > 1$$
log(9/5) -------- /231\ > 1 log|---| \100/
Then
$$x < -1$$
no execute
one of the solutions of our inequality is:
$$x > -1 \wedge x < 4$$
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x1 x2
Solving inequality on a graph
Rapid solution
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/ / ___\ / ___ \\ | | 1 \/ 5 | | 1 \/ 5 || Or|And|-1 < x, x < - - -----|, And|x < 4, - + ----- < x|| \ \ 2 2 / \ 2 2 //
$$\left(-1 < x \wedge x < \frac{1}{2} - \frac{\sqrt{5}}{2}\right) \vee \left(x < 4 \wedge \frac{1}{2} + \frac{\sqrt{5}}{2} < x\right)$$
((-1 < x)∧(x < 1/2 - sqrt(5)/2))∨((x < 4)∧(1/2 + sqrt(5)/2 < x))
Rapid solution 2
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___ ___
1 \/ 5 1 \/ 5
(-1, - - -----) U (- + -----, 4)
2 2 2 2
$$x\ in\ \left(-1, \frac{1}{2} - \frac{\sqrt{5}}{2}\right) \cup \left(\frac{1}{2} + \frac{\sqrt{5}}{2}, 4\right)$$
x in Union(Interval.open(-1, 1/2 - sqrt(5)/2), Interval.open(1/2 + sqrt(5)/2, 4))