document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150847>Question 1150847</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve Inequality: 2y^2 < y+1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772376 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=772376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Answer: <font color=red size=4>-1/2 < y < 1</font>\r<BR>\n" );
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document.write( "In interval notation, that would look like <font color=red size=4>(-1/2, 1)</font>. <BR>\n" );
document.write( "Make sure to not confuse this with ordered pair notation which unfortunately looks identical to interval notation if you use curved parenthesis for both endpoints.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "2y^2 < y+1<BR>\n" );
document.write( "2y^2 - y < 1<BR>\n" );
document.write( "2y^2 - y - 1  < 0<BR>\n" );
document.write( "2y^2 - 2y + y - 1  < 0<BR>\n" );
document.write( "(2y^2 - 2y) + (y - 1)  < 0<BR>\n" );
document.write( "2y(y - 1) + 1(y - 1)  < 0<BR>\n" );
document.write( "(2y + 1)(y - 1)  < 0\r<BR>\n" );
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document.write( "Solve (2y + 1)(y - 1) = 0 to get<BR>\n" );
document.write( "(2y + 1)(y - 1) = 0<BR>\n" );
document.write( "2y+1 = 0 or y-1 = 0<BR>\n" );
document.write( "2y = -1 or y = 1<BR>\n" );
document.write( "y = -1/2 or y = 1\r<BR>\n" );
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document.write( "These roots will help set up the boundaries of the intervals. Draw out a number line and plot the points -1/2 and 1 on the number line<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/M6ADAVY.png\"><BR>\n" );
document.write( "Note how I have color-coded three regions A, B, C<BR>\n" );
document.write( "Region A = interval to the left of -1/2<BR>\n" );
document.write( "Region B = interval between -1/2 and 1<BR>\n" );
document.write( "Region C = interval to the right of 1\r<BR>\n" );
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document.write( "Set up a sign chart as shown below. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/M5x6ypk.png\"><BR>\n" );
document.write( "The way I filled this chart out was plugging the representative y values for each region into each factor, then noting the sign (positive or negative). For instance, plug in y = -1 into each factor (2y+1) and (y-1) and you should get a negative result. We dont care what the actual value is. All we need is the sign of the results. When you multiply two negative values, you get a positive value. So thats why (2y + 1)(y - 1) is positive when -infinity < y < -1/2, or simply when y < -1/2. So region A is not part of the final answer. The other cells of the table are filled out in a similar way. I've highlighted the only region in which we get a negative value which is region B.\r<BR>\n" );
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document.write( "So this is why the answer is -1/2 < y < 1\r<BR>\n" );
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document.write( "A similar problem is solving <BR>\n" );
document.write( "2x^2 < x+1<BR>\n" );
document.write( "which turns into<BR>\n" );
document.write( "2x^2-x-1 < 0<BR>\n" );
document.write( "Let f(x) = 2x^2-x-1<BR>\n" );
document.write( "We want to find x values such that f(x) < 0<BR>\n" );
document.write( "We can find them through graphing<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/3qzcayX.png\"><BR>\n" );
document.write( "The region in blue represents the solution set, which is from x = -1/2 to x = 1, excluding both endpoints. In this blue region, the red f(x) curve is below the x axis. <BR>\n" );
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