document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=411562>Question 411562</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the values of the variable for which the expression is defined as a real number.\r<BR>\n" );
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document.write( "square root of 16-9x^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #289490 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=289490\">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=sqrt%2816-9x%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%2816-9x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In order for an even-numbered root (square root, 4th root, 6th root, etc.) to be a real nnumber, the radicand (the expression inside the radical) must be non-negative (positive or zero). So in order for your square root to be real, the radicand, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-9x%5E2 ALIGN=MIDDLE  ALT=\"16-9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, must be non-negative. In other \"words\":<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-9x%5E2+%3E=+0 ALIGN=MIDDLE  ALT=\"16-9x%5E2+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a quadratic inequality. These can be solved graphically or algebraically.<br><BR>\n" );
document.write( "Graphical solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-9x%5E2+%3E=+0 ALIGN=MIDDLE  ALT=\"16-9x%5E2+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Consider the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+16-5x%5E2 ALIGN=MIDDLE  ALT=\"y+=+16-5x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If know your equations you will recognize this as a parabola (because of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > but no <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) which opens downward (because of the minus in front of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term). The part of this parabola from the x-axis and up (if any) will have y coordinates that are zero or positive. The x values for this part of the parabola represent the solution to your problem because<ul><li>those x's make y zero or positive</li><li>Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++16-4x%5E2 ALIGN=MIDDLE  ALT=\"y+=++16-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, those same x's will make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-4x%5E2 ALIGN=MIDDLE  ALT=\"16-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > zero or positive.</li><li>We are looking for the x's that make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-4x%5E2 ALIGN=MIDDLE  ALT=\"16-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > zero or positive.</li></ul><BR>\n" );
document.write( "Here's a graph to help you see this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C+400%2C+-20%2C+20%2C+-20%2C+20%2C+16-4x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C+400%2C+-20%2C+20%2C+-20%2C+20%2C+16-4x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We are looking for the x's in that \"bump\" from the x-axis and up. The x's that make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-4x%5E2 ALIGN=MIDDLE  ALT=\"16-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > zero will be where the \"bump\" intersects the x-axis. We want these x values and everything in between.<br><BR>\n" );
document.write( "To find the x-intercepts we set y to zero (because y is zero on the x-axis) and solve the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-x%5E2+=+0 ALIGN=MIDDLE  ALT=\"16-x%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring we get:<BR>\n" );
document.write( "(4+3x)(4-3x) = 0<BR>\n" );
document.write( "From the Zero Product Property we know that one of these factors must be zero:<BR>\n" );
document.write( "4+3x = 0 or 4-3x=0<BR>\n" );
document.write( "Solving these we get:<BR>\n" );
document.write( "x = -4/3 or x = 4/3<BR>\n" );
document.write( "So the solution to your problem is all x's between -4/3 and 4/3, inclusive. In algebraic notation this is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3+%3C=+x+%3C=+4%2F3 ALIGN=MIDDLE  ALT=\"-4%2F3+%3C=+x+%3C=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "An algebraic solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-9x%5E2+%3E=+0 ALIGN=MIDDLE  ALT=\"16-9x%5E2+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First we factor:<BR>\n" );
document.write( "(4+3x)(4-3x) >= 0<BR>\n" );
document.write( "We have a product that is greater than or equal to zero. With a little thought we can figure out there are two ways this could happen:<ul><li>Both factors are positive (or zero)</li><li>Both factors are negative (or zero)</li></ul><BR>\n" );
document.write( "Algebraically \"both factors are positive (or zero)\" is expressed as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B3x+%3E=+0+ ALIGN=MIDDLE  ALT=\"4%2B3x+%3E=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3x+%3E=+0 ALIGN=MIDDLE  ALT=\"4-3x+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Algebraically \"both factors are negative (or zero)\" is expressed as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B3x+%3C=+0+ ALIGN=MIDDLE  ALT=\"4%2B3x+%3C=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3x+%3C=+0 ALIGN=MIDDLE  ALT=\"4-3x+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And finally to \"say\" that one or the other of these is true would be:<BR>\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B3x+%3E=+0+ ALIGN=MIDDLE  ALT=\"4%2B3x+%3E=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3x+%3E=+0 ALIGN=MIDDLE  ALT=\"4-3x+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) or (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B3x+%3C=+0+ ALIGN=MIDDLE  ALT=\"4%2B3x+%3C=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3x+%3C=+0 ALIGN=MIDDLE  ALT=\"4-3x+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "We can solve this compound inequality. Solving each of the individual inequalities 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=+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+%3C=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+%3C=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" >)<BR>\n" );
document.write( "(Note: If you cannot see how the 2nd and 4th inequalities work out the way they do, see below.)<BR>\n" );
document.write( "The first pair can be expressed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3+%3C=+x+%3C=+4%2F3 ALIGN=MIDDLE  ALT=\"-4%2F3+%3C=+x+%3C=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the same answer we got in the graphical solution. The second pair has NO solution because it is impossible for x to be less than or equal to -4/3 and greater than or equal to 4/3 <i>at the same time!</i><br><BR>\n" );
document.write( "So either graphically or algebraically the solution is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3+%3C=+x+%3C=+4%2F3 ALIGN=MIDDLE  ALT=\"-4%2F3+%3C=+x+%3C=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If you had trouble solving the 4-3x inequalities....<BR>\n" );
document.write( "Because of the minus in front of the x these are a little tricky to solve. For example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3x+%3E=+0 ALIGN=MIDDLE  ALT=\"4-3x+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Most people would start by subtracting 4 from each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x+%3E=+-4 ALIGN=MIDDLE  ALT=\"-3x+%3E=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then dividing by -3. But here is the \"trick\". We have to remember the special rule that applies whenever you multiply or divide both sides of an inequality by any negative number: You must reverse the inequality symbol! So when we divide by -3 the inequality gets reversed:<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;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );