document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=411433>Question 411433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve equation and inequality. Use set builder notation or interval notation to write solution sets to the inequalities.\r<BR>\n" );
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document.write( "(a) 18x^2 + 9x - 20 = 0\r<BR>\n" );
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document.write( "(b) 18x^2 + 9x - 20 (-/=)0\r<BR>\n" );
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document.write( "(c) 18x^2 + 9x - 20 (+/=)0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #289734 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=289734\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x%5E2%2B9x-20+=+0 ALIGN=MIDDLE  ALT=\"18x%5E2%2B9x-20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To solve this equation we factor it (or use the Quadratic Formula). Factoring it we get:<BR>\n" );
document.write( "(6x-5)(3x+4) = 0<BR>\n" );
document.write( "From the Zero Product Property we know that one of these factors must be zero. So:<BR>\n" );
document.write( "6x-5 = 0 or 4x+3 = 0<BR>\n" );
document.write( "Solving each of these we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%2F6 ALIGN=MIDDLE  ALT=\"x+=+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To solve the inequalities we will use the answers to the equation. When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%2F6 ALIGN=MIDDLE  ALT=\"x+=+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > the factor (6x-5) is zero. When x is greater than 5/6, the factor will be larger than zero. (IOW: positive). And when x is less than 5/6 the factor willbe smaller than zero (negative). Using the same logic on the other factor we find that when x is larger than -4/3 the (4x+3) factor will be positive and when x is smaller than -4/3, that factor will be negative.<br><BR>\n" );
document.write( "Now let's put this together. Think of a number line with the numbers -4/3 and 5/6 plotted. These two points divide the number line into three parts:<ul><li>The part to the left of -4/3</li><li>The part to the right of 5/6</li><li>The part between -4/3 and 5/6</li></ul><BR>\n" );
document.write( "Let's analyze what each factor, (6x-5) and (4x+3), will be in each of these parts of the number line:<ul><li>For x's to the left of -4/3, both factors will be negative. And since a negative times a negative is positive, (6x-5)(4x+3) will be positive.</li><li>For x's to the right of 5/6, both factors will be positive. And since a positive times a positive is positive, (6x-5)(4x+3) will be positive.</li><li>In between -4/3 and 5/6, (4x+3) will be positive and (6x-5) will be negative. And since a positive time a negative is negative, (6x-5)(4x+3) will be negative.</li></ul><BR>\n" );
document.write( "So we have found that if<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+-4%2F3 ALIGN=MIDDLE  ALT=\"x+%3C+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+6%2F5 ALIGN=MIDDLE  ALT=\"x+%3E+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then (6x-5)(4x+3) will be positive. If we include the values that make the factors zero we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+%3C=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+6%2F5 ALIGN=MIDDLE  ALT=\"x+%3E=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x%5E2+%2B+9x+-+20+%3E=+0 ALIGN=MIDDLE  ALT=\"18x%5E2+%2B+9x+-+20+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In set notation this is {x | <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+%3C=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+6%2F5 ALIGN=MIDDLE  ALT=\"x+%3E=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >}<br><BR>\n" );
document.write( "And if<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+-4%2F3 ALIGN=MIDDLE  ALT=\"x+%3E+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+5%2F6 ALIGN=MIDDLE  ALT=\"x+%3C+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then (6x-5)(4x+3) will be negative. If we include the values that make the factors zero we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+%3E=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+5%2F6 ALIGN=MIDDLE  ALT=\"x+%3C=+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x%5E2+%2B+9x+-+20+%3C=+0 ALIGN=MIDDLE  ALT=\"18x%5E2+%2B+9x+-+20+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In set notation this is {x | <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+%3E=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+5%2F6 ALIGN=MIDDLE  ALT=\"x+%3C=+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >} or<BR>\n" );
document.write( "{x | <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3+%3C=+x+%3C=+5%2F6 ALIGN=MIDDLE  ALT=\"-4%2F3+%3C=+x+%3C=+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >} </SPAN>\n" );
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