Question 247645
Here, we see two numbers are being multiplied together in the inequality ({{{x^2}}} and {{{(x+8)}}}).  And we know that the only way to multiply two numbers together and get a number less than 0 is to multiply a positive and a negative number together.  (positive * positive = positive; and negative * negative = positive)
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Thus, we know that {{{x^2(x+8)<0}}} is true in either of the following situations:
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1. {{{x^2 < 0}}} and {{{(x+8) > 0}}}
2. {{{x^2 > 0}}} and {{{(x+8) > 0}}}
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If you look at the first statement carefully, you may note that it can never be true.  As previously stated, positive * positive = positive; and negative * negative = positive, so there is no way you can take any integer, square it, and get a negative number.  This means, we only need to concern ourselves with the second situation, {{{x^2 > 0}}} and {{{(x+8) > 0}}}.

So, let's solve the two inequalities:
{{{x^2 > 0}}} 
{{{x > 0}}}  (get the square root of both sides)
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 {{{(x+8) > 0}}}
{{{x > -8}}}
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So, we get <b>{{{x > -8}}}</b> as our answer.  (because all items > -8 are also > 0, so we use the more restrictive option as our final answer to ensure all possibilities work.)
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We can always double check our work.  Let's take {{{x = -10}}}, as that satisfies our answer condition.  Plugging it into our original inequality:
{{{(-10)^2(-10+8) <  0}}}
{{{100(-2) <  0}}}
{{{-200 <  0}}}, which is a true statement.