document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=695850>Question 695850</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let E(x) = 30(1.05)^x and L(x) = 30 + 2x.<BR>\n" );
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document.write( "I need to give an example of a value x when L(x) < E(x). Can someone please help? I am really struggling with this one. I have the rest of the problem figured out. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #428756 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=428756\">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=L%28x%29+=+30+%2B+2x ALIGN=MIDDLE  ALT=\"L%28x%29+=+30+%2B+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a cleverly named <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28L%29 ALIGN=MIDDLE  ALT=\"highlight%28L%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >inear function, while<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28x%29+=+30%281.05%29%5Ex ALIGN=MIDDLE  ALT=\"E%28x%29+=+30%281.05%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a cleverly named <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28E%29 ALIGN=MIDDLE  ALT=\"highlight%28E%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >xponential function.<BR>\n" );
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document.write( "Linear functions graph as straight lines, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28x%29+=+30+%2B+2x ALIGN=MIDDLE  ALT=\"L%28x%29+=+30+%2B+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents a line with a slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > throughout its all real numbers domain.<BR>\n" );
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document.write( "Exponential functions graph as curved lines, and if the base is greater than one, as for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28x%29+=+30%281.05%29%5Ex ALIGN=MIDDLE  ALT=\"E%28x%29+=+30%281.05%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >, have \"exponential growth\" graphs that look like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.2%2A2%5E%282x%29%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.2%2A2%5E%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with slopes that increase from very small to very large.<BR>\n" );
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document.write( "The line could be tangent to the exponential curve and the graphs could intersect at only one point, as in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.6x%2B0.18%2C0.2%2Ae%5E%282x%29%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.6x%2B0.18%2C0.2%2Ae%5E%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Otherwise, a linear curve and an exponential curve could intersect at two points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.3x%2B0.3%2C0.2%2A2%5E%282x%29%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.3x%2B0.3%2C0.2%2A2%5E%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or zero points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.3x%2B0.1%2C0.2%2A2%5E%282x%29%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.99%2C0.99%2C-0.99%2C0.99%2C0.3x%2B0.1%2C0.2%2A2%5E%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "In any of those cases, the line is below the exponential curve (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28x%29+%3C+E%28x%29 ALIGN=MIDDLE  ALT=\"L%28x%29+%3C+E%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) at most (if not all) points, for most (if not all) values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Which of those scenarios is the case for your functions?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%280%29=30%2B2%2A0=30%2B0=30 ALIGN=MIDDLE  ALT=\"L%280%29=30%2B2%2A0=30%2B0=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%280%29=30%281.05%29%5E0=30%2A1=30 ALIGN=MIDDLE  ALT=\"E%280%29=30%281.05%29%5E0=30%2A1=30\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the graphs intersect at a least one point, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We can look at points to either side.<BR>\n" );
document.write( "If the curves are tangent at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, for all other values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "it will be (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28x%29+%3C+E%28x%29 ALIGN=MIDDLE  ALT=\"L%28x%29+%3C+E%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "If the curves are not tangent at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then they intersect at two points,<BR>\n" );
document.write( "In that case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28x%29%3CE%28x%29 ALIGN=MIDDLE  ALT=\"L%28x%29%3CE%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be true at least in some interval to one side of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we may be lucky enough to find a point there.<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%281%29=30%2B2%2A1=30%2B2=32 ALIGN=MIDDLE  ALT=\"L%281%29=30%2B2%2A1=30%2B2=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%281%29=30%281.05%29%5E1=30%2A1.05=31.5 ALIGN=MIDDLE  ALT=\"E%281%29=30%281.05%29%5E1=30%2A1.05=31.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > so for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28x%29%3EE%28x%29 ALIGN=MIDDLE  ALT=\"L%28x%29%3EE%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28-1%29=30%2B2%2A%28-1%29=30-2=28 ALIGN=MIDDLE  ALT=\"L%28-1%29=30%2B2%2A%28-1%29=30-2=28\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%281%29=30%281.05%29%5E%28-1%29=30%2F1.05=200%2F7 ALIGN=MIDDLE  ALT=\"E%281%29=30%281.05%29%5E%28-1%29=30%2F1.05=200%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >=approx.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28.57 ALIGN=MIDDLE  ALT=\"28.57\"   DISABLED_style= \"max-width:90%; height:auto;\" > so for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-1%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28x%29%3CE%28x%29 ALIGN=MIDDLE  ALT=\"L%28x%29%3CE%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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