document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=478755>Question 478755</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone please help solve this equation<BR>\n" );
document.write( "6x(x-2)(3x+5)<0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #328100 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=328100\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> easy way to solve it is to graph it.<BR>\n" );
document.write( "that's kind of like cheating, but here goes.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-3%2C3%2C-100%2C100%2C6x%2A%28x-2%29%2A%283x%2B5%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-3%2C3%2C-100%2C100%2C6x%2A%28x-2%29%2A%283x%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "by looking at the graph you can see the regions that it's positive in and the regions that it's negative in.<BR>\n" );
document.write( "if you want to do it by formula, then you would want to find the 0 points of equation.<BR>\n" );
document.write( "set the equation equal to 0 to get:<BR>\n" );
document.write( "6x(x-2)(3x+5)=0<BR>\n" );
document.write( "this equation will be 0 if:<BR>\n" );
document.write( "6x = 0<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x-2 = 0<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "3x+5 = 0<BR>\n" );
document.write( "solve for x in each of those equations and you get the overall equation being when:<BR>\n" );
document.write( "x = 0<BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "x = -5/3<BR>\n" );
document.write( "look at the graph and you'll see that's where the graph crosses the x-axis.<BR>\n" );
document.write( "if you didn't have a graph to look at, you would simply know that the equation is equal to 0 when x is the values shown.<BR>\n" );
document.write( "the equation you are using is a continuous function.<BR>\n" );
document.write( "this means that there are no breaks in the line of the equation.<BR>\n" );
document.write( "it is smooth and continuous from end to end.<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "before or after or in between the 0 points of the equation, if it's negative it will stay negative, and if it's positive it will stay positive.<BR>\n" );
document.write( "since you know the 0 points, just check each section before and after and between when x = 0 to see if the y value of the equation is negative or positive.<BR>\n" );
document.write( "pick a value for x and solve the equation using that value.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "assume x = -2 which is less than -5/3.<BR>\n" );
document.write( "when x = -2, the expression of 6x * (x-2) * (3x+5) = -12*(-4)*(-1) which will be negative because you have a minus times a minus times a minus.<BR>\n" );
document.write( "this means the equation is negative before the value of x = -5/3.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "assume x = -1 which is greater than -5/3 but less than 0.<BR>\n" );
document.write( "when x = -1, the expression of 6x * (x-2) * (3x+5) = -6*(-3)*2) = 36 which is a minus times a minus times a plus.<BR>\n" );
document.write( "this means that the equation is positive between the values of x = -5/3 and x = 0.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "assume x = 1 which is greater than x = 0 and less than x = 2.<BR>\n" );
document.write( "when x = 1, the expression of  6x * (x-2) * (3x+5) = 6*(-1)*(15) which is a plus times a minus times a plus.<BR>\n" );
document.write( "this means that the equation is negative between the values of x = 0 and x = 2.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "assume x = 3 which is greater than x = 2.<BR>\n" );
document.write( "when x = 3, the expression of  6x * (x-2) * (3x+5) = 18*1*14 which is a plus times a plus times a plus.<BR>\n" );
document.write( "this means that the equation is positive after the value of x = 2.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "put these all together and you get:<BR>\n" );
document.write( "when x < -5/3, the equation is negative.<BR>\n" );
document.write( "when x > -5/3 but < 0, the equation is positive<BR>\n" );
document.write( "when x > 0 but < 2, the equation is negative.<BR>\n" );
document.write( "when x > 2, the equation is positive.<BR>\n" );
document.write( "the question is to find when the equation is negative.<BR>\n" );
document.write( "that occurs when:\"<BR>\n" );
document.write( "x < -5/3<BR>\n" );
document.write( "x > 0 but < 2<BR>\n" );
document.write( "all other times the equation is either equal to 0 or greater than 0.<BR>\n" );
document.write( "this can be written as:<BR>\n" );
document.write( "x < -5/3 or 0 < x < 2<BR>\n" );
document.write( "check the graph again to see if it concurs.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );