document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=695832>Question 695832</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the solution of the system of inequalities?\r<BR>\n" );
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document.write( "y&#8805; x^2+2x+2<BR>\n" );
document.write( "y< -x^2-4x-2\r<BR>\n" );
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document.write( "Thank You! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #430120 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=430120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First let's look at the graph of the two parabolas:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2B2x%2B2%2C+-x%5E2-4x-2%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2B2x%2B2%2C+-x%5E2-4x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The red one is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2%2B2x%2B2 ALIGN=MIDDLE  ALT=\"y+=+x%5E2%2B2x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the green one is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-x%5E2-4x-2 ALIGN=MIDDLE  ALT=\"y+=+-x%5E2-4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+x%5E2%2B2x%2B2 ALIGN=MIDDLE  ALT=\"y+%3E=+x%5E2%2B2x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be the red parabola and all the area above/inside the bowl. The solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C+-x%5E2-4x-2 ALIGN=MIDDLE  ALT=\"y+%3C+-x%5E2-4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be all the area below/inside the green parabola. (Not the green parabola itself because that is where y equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2-4x-2 ALIGN=MIDDLE  ALT=\"-x%5E2-4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the inequality does not include \"or equal to\".)<br><BR>\n" );
document.write( "So the solution to the system is where these two areas overlap each other: The enclosed area between the two parabolas. To express this solution we must first find the two points where the parabolas intersect. Setting<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x%2B2+=+-x%5E2-4x-2 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x%2B2+=+-x%5E2-4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and solving for x we should be able to find the points of intersection. Subtracting the entire right side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B6x%2B4+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B6x%2B4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factor out the GCF of 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%5E2%2B3x%2B2%29+=+0 ALIGN=MIDDLE  ALT=\"2%28x%5E2%2B3x%2B2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring the trinomial:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2B1%29%28x%2B2%29+=+0 ALIGN=MIDDLE  ALT=\"2%28x%2B1%29%28x%2B2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From the Zero Product Property:<BR>\n" );
document.write( "x + 1 = 0 or x + 2 = 0<BR>\n" );
document.write( "Solving:<BR>\n" );
document.write( "x = -1 or x = -2<br><BR>\n" );
document.write( "Using these x values and either equation for the parabola we can find that the y values for each x are 1 and 2, respectively. So the points of intersection are:<BR>\n" );
document.write( "(-1, 1) and (-2, 2)<BR>\n" );
document.write( "So the x values of the solution are between -1 and -2, inclusive:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2+%3C+x+%3C+-1 ALIGN=MIDDLE  ALT=\"-2+%3C+x+%3C+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the y values of the solution are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x%2B2+%3C=+y+%3C+-x%5E2-4x-2 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x%2B2+%3C=+y+%3C+-x%5E2-4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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