document.write( "Question 1087724: Solve the quadratic inequality. Write the solution set in interval notation. Show the complete solution.
\n" ); document.write( " 𝑎2+3𝑎+2<−3(𝑎+2) \n" ); document.write( "

Algebra.Com's Answer #702048 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
I will write it as
\n" ); document.write( " $\"a%5E2%2B3a%2B2%3C-3%28a%2B2%29\"$ , because I am assuming that that is what you meant.
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\n" ); document.write( "To solve inequalities, we transform them into equivalent inequalities
\n" ); document.write( "(meaning inequalities that have exactly all the same solutions).
\n" ); document.write( "To do that it is always safe to add the same number or expression to both sides.
\n" ); document.write( "In this case, I found that factoring that quadratic polynomial from the start was easier for me.
\n" ); document.write( " $\"a%5E2%2B3a%2B2%3C-3%28a%2B2%29\"$
\n" ); document.write( " $\"%28a%2B1%29%28a%2B2%29%3C-3%28a%2B2%29\"$
\n" ); document.write( " $\"%28a%2B1%29%28a%2B2%29%2B3%28a%2B2%29%3C-3%28a%2B2%29%2B3%28a%2B2%29\"$
\n" ); document.write( " $\"%28%28a%2B1%29%2B3%29%28a%2B2%29%3C0\"$
\n" ); document.write( " $\"%28a%2B4%29%28a%2B2%29%3C0\"$
\n" ); document.write( "That last inequality could also be written as $\"a%5E2%2B6a%2B8%3C0\"$ ,
\n" ); document.write( "but that is not needed in this case.
\n" ); document.write( "In fact, the factored form makes it easier to see the solution.
\n" ); document.write( "You know that the zeros of $\"%28a%2B4%29%28a%2B2%29\"$ are $\"system%28a=-2%2C%22and%22%2Ca=-4%29\"$ ,
\n" ); document.write( "and that is where the sign changes for each factor, and for that quadratic polynomial.
\n" ); document.write( "In between those two zeros it is negative,
\n" ); document.write( "so the solution to $\"%28a%2B4%29%28a%2B2%29%3C0\"$ is
\n" ); document.write( " $\"-4%3Ca%3C-2\"$ , or $\"highlight%28%22%28+-4+%2C+-2+%29%22%29\"$ .
\n" ); document.write( " $\"graph%28300%2C300%2C-8%2C2%2C-4%2C16%2C%28x%2B2%29%28x%2B4%29%29\"$
\n" ); document.write( "That the quadratic polynomial is negative between -4 and -2 is obvioumany different ways.
\n" ); document.write( "Looking at the quadratic function/expression/polynomial $\"a%5E2%2B6a%2B8\"$ ,
\n" ); document.write( "you see that the leading coefficient is that invisible $\"1%3E0\"$
\n" ); document.write( "in the $\"a%5E2\"$ leading term ,
\n" ); document.write( "so its graph opens up, and the polynomial is negative between the roots.
\n" ); document.write( "It is also obvious because $\"%28a%2B4%29%28a%2B2%29\"$ is $\"2%2A4=8%3E0\"$ for $\"a=0\"$ ,
\n" ); document.write( "and since the polynomial changes sign for multiplicity 1 zeros a=-2, and, a=-4.
\n" ); document.write( "that would make it so that
\n" ); document.write( " $\"%28a%2B4%29%28a%2B2%29%3E0\"$ for $\"-2%3Ca\"$ (including for a=0),
\n" ); document.write( " $\"%28a%2B4%29%28a%2B2%29%3C0\"$ for $\"-4%3Ca%3C-2\"$ , and
\n" ); document.write( " $\"%28a%2B4%29%28a%2B2%29%3E0\"$ for $\"a%3C-4\"$ .
\n" ); document.write( "You could also look at the sign of each factor.
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