document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195834>Question 1195834</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many ordered pairs of integers (a,b) satisfy all of the following inequalities?<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 #828400 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828400\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "*** Note that another tutor has pointed out that my solution below overlooks<BR>\n" );
document.write( "*** two other solutions, (3,2) and (2,3) -- so there are 6 ordered pairs that<BR>\n" );
document.write( "*** satisfy the given inequalities, not just 4.<br><BR>\n" );
document.write( "First, notice that, by symmetry, if (m,n) satisfies all three conditions then (n,m) does also.<br><BR>\n" );
document.write( "Next, since a^2 and b^2 are non-negative, conditions 2 and 3 mean a and b are positive; and condition 1 means a and b are both less than 4.<br><BR>\n" );
document.write( "So we know a and b are both either 1, 2, or 3.  There aren't a lot of ordered pairs to look at, so a \"brute force\" solution (by simply trying all the ordered pairs with both coordinates 1, 2, or 3) will be easier than a formal algebraic solution.<br><BR>\n" );
document.write( "Trial and error show that (1,1) and (2,2) satisfy the conditions, but (3,3) violates condition 1.<br><BR>\n" );
document.write( "Trial and error shows that (2,1) satisfies the conditions, so (1,2) does also.  And similar trial and error shows that (3,1) and (1,3) do not satisfy all the conditions.<br><BR>\n" );
document.write( "So we have found all the ordered pairs of integers that satisfy all three conditions: (1,1), (2,2), (1,2), and (2,1).<br><BR>\n" );
document.write( "ANSWER: 4<br><BR>\n" );
document.write( "You can easily verify this answer by using x and y instead of a and b and graphing all three equations (not the inequalities) on desmos.com.  The ordered pairs that satisfy all three conditions are the lattice points that are inside all three circles.<br><BR>\n" );
document.write( "Of course, you could also find the answer by doing the graphing first, without the logical analysis I showed in the first part of my response.<br><BR>\n" );
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