Question

How do I solve this equation, x−5 /(x2+x+5)(x2−4x−5)>0?
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Answer 1

I hope you will get it.

Answer 2

Answer:

The solution of the given inequality is

$x∈( - 1, \: 5)U(5 \:, \infin)$

Step-by-step explanation:

The given inequality is

$\frac{x - 5}{(x {}^{2} + x + 5)( {x}^{2} - 4x - 5) } > 0$

In the denominator we have an equation

${x}^{2} + x + 5$

The discriminant of the equation is

$∆=1^{2}-4(1)(5)=-19<0$

From theory of equations we know that if discriminant if equation is zero then

$∆<0 \\ = > a {x}^{2} + bx + c > 0(a > 0)$

Therefore we have

${x}^{2} + x + 5 > 0$

Now we can ignore this equation and focus on the left over part which is

$\frac{x - 5}{ {x}^{2} - 4x - 5 } > 0 \\ = > \frac{x - 5}{(x - 5)(x + 1)} > 0 \\ = > \frac{1}{x + 1} > 0 \:(x≠5) \\ = > x > - 1$

Therefore the conditions established are

$x > - 1 \: and \: x≠5 \\ = > x∈( - 1, \: 5)U(5 \:, \infin)$

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