document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209632>Question 1209632</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the area of the region of the complex plane defined by |z| < 3 + 2|z - 1|? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849732 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to approach finding the area of the region:\r<BR>\n" );
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document.write( "1. **Rewrite in terms of x and y:**\r<BR>\n" );
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document.write( "Let z = x + yi, where x and y are real numbers. Then the inequality becomes:\r<BR>\n" );
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document.write( "|x + yi| < 3 + 2|x + yi - 1|<BR>\n" );
document.write( "√(x² + y²) < 3 + 2√((x - 1)² + y²)\r<BR>\n" );
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document.write( "2. **Consider the cases:**\r<BR>\n" );
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document.write( "This inequality isn't easy to work with directly.  Squaring both sides will remove the square roots, but it will still be messy.  Geometrically, |z| represents the distance from the origin, and |z - 1| represents the distance from the point (1, 0).  The inequality says the distance from the origin is less than 3 plus twice the distance from (1, 0).\r<BR>\n" );
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document.write( "3. **Try some test points:**\r<BR>\n" );
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document.write( "* z = 0: |0| < 3 + 2|0 - 1|  => 0 < 3 + 2, which is true.<BR>\n" );
document.write( "* z = 1: |1| < 3 + 2|1 - 1|  => 1 < 3, which is true.<BR>\n" );
document.write( "* z = -1: |-1| < 3 + 2|-1 - 1| => 1 < 3 + 4, which is true.<BR>\n" );
document.write( "* z = i: |i| < 3 + 2|i - 1| => 1 < 3 + 2√2, which is true.\r<BR>\n" );
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document.write( "4. **Consider the boundary:**\r<BR>\n" );
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document.write( "The boundary of the region is given by the equation:\r<BR>\n" );
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document.write( "√(x² + y²) = 3 + 2√((x - 1)² + y²)\r<BR>\n" );
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document.write( "Squaring both sides:<BR>\n" );
document.write( "x² + y² = 9 + 12√((x - 1)² + y²) + 4((x - 1)² + y²)<BR>\n" );
document.write( "x² + y² = 9 + 12√((x - 1)² + y²) + 4(x² - 2x + 1 + y²)<BR>\n" );
document.write( "x² + y² = 9 + 12√((x - 1)² + y²) + 4x² - 8x + 4 + 4y²<BR>\n" );
document.write( "-3x² + 8x - 13 - 3y² = 12√((x - 1)² + y²)\r<BR>\n" );
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document.write( "Squaring again would lead to a quartic equation, which is generally quite difficult to solve.\r<BR>\n" );
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document.write( "5. **Numerical/Graphical Approach:**\r<BR>\n" );
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document.write( "Given the difficulty of the algebra, a numerical or graphical approach is recommended.  Plot the boundary equation (or inequality) using software like Wolfram Alpha, Desmos, or similar tools. This will give a visual representation of the region.\r<BR>\n" );
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document.write( "6. **Approximation:**\r<BR>\n" );
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document.write( "The region appears to be a somewhat distorted circle.  We can try to estimate the area.  The inequality implies that the region is contained within a circle of radius somewhat larger than 3 centered at the origin.\r<BR>\n" );
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document.write( "The area is difficult to calculate analytically.  A numerical or graphical approach is the most practical way to determine the area.<BR>\n" );
document.write( " </SPAN>\n" );
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