Mister Exam
log2(x^2+4x+3)>0 inequation
The solution
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[src]
/ 2 \
log\x + 4*x + 3/
----------------- > 0
log(2)
$$\frac{\log{\left(\left(x^{2} + 4 x\right) + 3 \right)}}{\log{\left(2 \right)}} > 0$$
log(x^2 + 4*x + 3)/log(2) > 0
Detail solution
Given the inequality:
$$\frac{\log{\left(\left(x^{2} + 4 x\right) + 3 \right)}}{\log{\left(2 \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(\left(x^{2} + 4 x\right) + 3 \right)}}{\log{\left(2 \right)}} = 0$$
Solve:
$$x_{1} = -2 - \sqrt{2}$$
$$x_{2} = -2 + \sqrt{2}$$
$$x_{1} = -2 - \sqrt{2}$$
$$x_{2} = -2 + \sqrt{2}$$
This roots
$$x_{1} = -2 - \sqrt{2}$$
$$x_{2} = -2 + \sqrt{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(-2 - \sqrt{2}\right) + - \frac{1}{10}$$
=
$$- \frac{21}{10} - \sqrt{2}$$
substitute to the expression
$$\frac{\log{\left(\left(x^{2} + 4 x\right) + 3 \right)}}{\log{\left(2 \right)}} > 0$$
$$\frac{\log{\left(\left(4 \left(- \frac{21}{10} - \sqrt{2}\right) + \left(- \frac{21}{10} - \sqrt{2}\right)^{2}\right) + 3 \right)}}{\log{\left(2 \right)}} > 0$$
one of the solutions of our inequality is:
$$x < -2 - \sqrt{2}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -2 - \sqrt{2}$$
$$x > -2 + \sqrt{2}$$
$$\frac{\log{\left(\left(x^{2} + 4 x\right) + 3 \right)}}{\log{\left(2 \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(\left(x^{2} + 4 x\right) + 3 \right)}}{\log{\left(2 \right)}} = 0$$
Solve:
$$x_{1} = -2 - \sqrt{2}$$
$$x_{2} = -2 + \sqrt{2}$$
$$x_{1} = -2 - \sqrt{2}$$
$$x_{2} = -2 + \sqrt{2}$$
This roots
$$x_{1} = -2 - \sqrt{2}$$
$$x_{2} = -2 + \sqrt{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(-2 - \sqrt{2}\right) + - \frac{1}{10}$$
=
$$- \frac{21}{10} - \sqrt{2}$$
substitute to the expression
$$\frac{\log{\left(\left(x^{2} + 4 x\right) + 3 \right)}}{\log{\left(2 \right)}} > 0$$
$$\frac{\log{\left(\left(4 \left(- \frac{21}{10} - \sqrt{2}\right) + \left(- \frac{21}{10} - \sqrt{2}\right)^{2}\right) + 3 \right)}}{\log{\left(2 \right)}} > 0$$
/ 2 \
| 27 / 21 ___\ ___|
log|- -- + |- -- - \/ 2 | - 4*\/ 2 |
\ 5 \ 10 / / > 0
-------------------------------------
log(2)
one of the solutions of our inequality is:
$$x < -2 - \sqrt{2}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -2 - \sqrt{2}$$
$$x > -2 + \sqrt{2}$$
Solving inequality on a graph
Rapid solution
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/ / ___\ ___ \ Or\And\-oo < x, x < -2 - \/ 2 /, -2 + \/ 2 < x/
$$\left(-\infty < x \wedge x < -2 - \sqrt{2}\right) \vee -2 + \sqrt{2} < x$$
(-2 + sqrt(2) < x)∨((-oo < x)∧(x < -2 - sqrt(2)))
Rapid solution 2
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___ ___ (-oo, -2 - \/ 2 ) U (-2 + \/ 2 , oo)
$$x\ in\ \left(-\infty, -2 - \sqrt{2}\right) \cup \left(-2 + \sqrt{2}, \infty\right)$$
x in Union(Interval.open(-oo, -2 - sqrt(2)), Interval.open(-2 + sqrt(2), oo))