document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208931>Question 1208931</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that: if a > 0, b > 0, and <BR>\n" );
document.write( "sqrt{a} < sqrt{b}, then a < b.\r<BR>\n" );
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document.write( "Hint given: \r<BR>\n" );
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document.write( "b - a = (sqrt{b} - sqrt{a})(sqrt{b} + sqrt{a}\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847460 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=847460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "sqrt(a) < sqrt(b)<BR>\n" );
document.write( "sqrt(b) > sqrt(a)<BR>\n" );
document.write( "sqrt(b)-sqrt(a) > 0<BR>\n" );
document.write( "(sqrt(b)-sqrt(a))*(sqrt(b)+sqrt(a)) > 0*(sqrt(b)+sqrt(a)) ....... see note below<BR>\n" );
document.write( "b-a > 0<BR>\n" );
document.write( "b > a<BR>\n" );
document.write( "a < b\r<BR>\n" );
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document.write( "Note:<BR>\n" );
document.write( "I multiplied both sides by (sqrt(b)+sqrt(a)) to use the hint your teacher gave you.<BR>\n" );
document.write( "The output of a square root function is never negative as long as there isn't a negative out front. Eg: sqrt(25) = 5. This means sqrt(a) and sqrt(b) are both positive. Furthermore it means sqrt(b)+sqrt(a) > 0. Multiplying both sides by a positive value will not flip the inequality sign.\r<BR>\n" );
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document.write( "Another approach.\r<BR>\n" );
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document.write( "sqrt(a) < sqrt(b) means that <font color=blue>sqrt(a) = sqrt(b) - c</font> for some positive real number c.<BR>\n" );
document.write( "In other words, sqrt(a) = sqrt(b) - somePositiveNumber\r<BR>\n" );
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document.write( "Square both sides of this equation.<BR>\n" );
document.write( "sqrt(a) = sqrt(b) - c<BR>\n" );
document.write( "( sqrt(a) )^2 = ( sqrt(b) - c)^2<BR>\n" );
document.write( "a = ( sqrt(b) )^2 - 2*c*sqrt(b) + c^2<BR>\n" );
document.write( "a = b - 2*c*sqrt(b) + c^2<BR>\n" );
document.write( "a = b - (2*c*sqrt(b) - c^2)<BR>\n" );
document.write( "a = b - c*(2*sqrt(b) - c)<BR>\n" );
document.write( "a = b - c*(sqrt(b) + <font color=blue>sqrt(b) - c</font>)<BR>\n" );
document.write( "a = b - c*(sqrt(b) + <font color=blue>sqrt(a)</font>)<BR>\n" );
document.write( "a = b - positiveNumber*(positiveNumber + positiveNumber)<BR>\n" );
document.write( "a = b - positiveNumber*positiveNumber<BR>\n" );
document.write( "a = b - positiveNumber<BR>\n" );
document.write( "a < b\r<BR>\n" );
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document.write( "In the 3rd and 4th steps, notice that the squaring and square roots cancel each other out.<BR>\n" );
document.write( "For instance ( sqrt(b) )^2 = b when b > 0.<BR>\n" );
document.write( "The terms marked in blue represent a substitution based on the initial equation. <BR>\n" );
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