document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194429>Question 1194429</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The numerator of the fraction is x, while the denominator is 5 more the numerator if the value of the fraction is less than 2/5, find the value of x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #826629 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=826629\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = numerator<BR>\n" );
document.write( "x+5 = denominator\r<BR>\n" );
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document.write( "x/(x+5) is the fraction itself<BR>\n" );
document.write( "It is stated to be less than 2/5\r<BR>\n" );
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document.write( "x/(x+5) < 2/5\r<BR>\n" );
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document.write( "Notice that x = -5 leads to a division by zero error.<BR>\n" );
document.write( "This is a natural cutting point to divide the x axis number line into two pieces, and we'll test each piece.\r<BR>\n" );
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document.write( "Let's plug in something to the left of x = -5, let's say x = -6<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "x/(x+5) < 2/5<BR>\n" );
document.write( "-6/(-6+5) < 2/5<BR>\n" );
document.write( "6 < 2/5<BR>\n" );
document.write( "6 < 0.4<BR>\n" );
document.write( "which is clearly false<BR>\n" );
document.write( "You'll find a false statement results when you plug in anything to the left of x = -5<BR>\n" );
document.write( "So we ignore x < -5 and x = -5 itself.\r<BR>\n" );
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document.write( "Let's try something to the right of x = -5. I'll try x = 0<BR>\n" );
document.write( "x/(x+5) < 2/5<BR>\n" );
document.write( "0/(0+5) < 2/5<BR>\n" );
document.write( "0 < 2/5<BR>\n" );
document.write( "0 < 0.4<BR>\n" );
document.write( "That is true. This suggests that x > -5 is the solution set.\r<BR>\n" );
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document.write( "However, we need to solve for x like so<BR>\n" );
document.write( "x/(x+5) < 2/5<BR>\n" );
document.write( "5x < 2(x+5)<BR>\n" );
document.write( "5x < 2x+10<BR>\n" );
document.write( "5x-2x < 10<BR>\n" );
document.write( "3x < 10<BR>\n" );
document.write( "x < 10/3<BR>\n" );
document.write( "The inequality sign stays the same the entire time. It only flips if we were to multiply or divide both sides by a negative number.\r<BR>\n" );
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document.write( "The reality is that yes x > -5 must be the case, but we also need x < 10/3 as well<BR>\n" );
document.write( "Put together, the solution set is -5 < x < 10/3\r<BR>\n" );
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document.write( "A quick way to verify this is to graph these equations<BR>\n" );
document.write( "y = x/(x+5)<BR>\n" );
document.write( "y = 2/5<BR>\n" );
document.write( "Then notice how the hyperbolic curve is below the horizontal line along the interval -5 < x < 10/3\r<BR>\n" );
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document.write( "Graph:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/fmus5i4ao0\">https://www.desmos.com/calculator/fmus5i4ao0</a><BR>\n" );
document.write( "Desmos is a free graphing app.<BR>\n" );
document.write( "You can click on the intersection of the horizontal line and the curve to see the coordinates of said intersection point.<BR>\n" );
document.write( "Keep in mind that 10/3 = 3.333 approximately.\r<BR>\n" );
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document.write( "Answer: Anything on the interval -5 < x < 10/3<BR>\n" );
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