document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209886>Question 1209886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the area of the region determined by the system<BR>\n" );
document.write( "y \ge |x|, \\<BR>\n" );
document.write( "y \le -|2x + 1| + 6. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850958 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=850958\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the inequalities to determine the region.\r<BR>\n" );
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document.write( "1.  **$y \ge |x|$**\r<BR>\n" );
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document.write( "    * This represents the region above the V-shaped graph of $y = |x|$.<BR>\n" );
document.write( "    * $y = x$ for $x \ge 0$<BR>\n" );
document.write( "    * $y = -x$ for $x < 0$\r<BR>\n" );
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document.write( "2.  **$y \le -|2x + 1| + 6$**\r<BR>\n" );
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document.write( "    * This represents the region below the inverted V-shaped graph of $y = -|2x + 1| + 6$.<BR>\n" );
document.write( "    * To find the vertex of $y = -|2x + 1| + 6$, we set $2x + 1 = 0$, which gives $x = -\frac{1}{2}$.<BR>\n" );
document.write( "    * When $x = -\frac{1}{2}$, $y = -|0| + 6 = 6$. So, the vertex is $(-\frac{1}{2}, 6)$.<BR>\n" );
document.write( "    * For $2x + 1 \ge 0$, i.e., $x \ge -\frac{1}{2}$, $y = -(2x + 1) + 6 = -2x + 5$.<BR>\n" );
document.write( "    * For $2x + 1 < 0$, i.e., $x < -\frac{1}{2}$, $y = -(-2x - 1) + 6 = 2x + 7$.\r<BR>\n" );
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document.write( "Now, we need to find the intersection points of the graphs.\r<BR>\n" );
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document.write( "**Intersection of $y = |x|$ and $y = -|2x + 1| + 6$:**\r<BR>\n" );
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document.write( "* **Case 1: $x \ge 0$ and $x \ge -\frac{1}{2}$ (i.e., $x \ge 0$)**<BR>\n" );
document.write( "    * $x = -2x + 5$<BR>\n" );
document.write( "    * $3x = 5$<BR>\n" );
document.write( "    * $x = \frac{5}{3}$<BR>\n" );
document.write( "    * $y = \frac{5}{3}$<BR>\n" );
document.write( "    * Intersection point: $(\frac{5}{3}, \frac{5}{3})$\r<BR>\n" );
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document.write( "* **Case 2: $x < 0$ and $x \ge -\frac{1}{2}$ (i.e., $-\frac{1}{2} \le x < 0$)**<BR>\n" );
document.write( "    * $-x = -2x + 5$<BR>\n" );
document.write( "    * $x = 5$ (This is not in the interval, so no intersection)\r<BR>\n" );
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document.write( "* **Case 3: $x \ge 0$ and $x < -\frac{1}{2}$ (Impossible)**\r<BR>\n" );
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document.write( "* **Case 4: $x < 0$ and $x < -\frac{1}{2}$ (i.e., $x < -\frac{1}{2}$) **<BR>\n" );
document.write( "    * $-x = 2x + 7$<BR>\n" );
document.write( "    * $-3x = 7$<BR>\n" );
document.write( "    * $x = -\frac{7}{3}$<BR>\n" );
document.write( "    * $y = \frac{7}{3}$<BR>\n" );
document.write( "    * Intersection point: $(-\frac{7}{3}, \frac{7}{3})$\r<BR>\n" );
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document.write( "The intersection points are $(\frac{5}{3}, \frac{5}{3})$ and $(-\frac{7}{3}, \frac{7}{3})$.\r<BR>\n" );
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document.write( "Now, we need to find the area of the region.\r<BR>\n" );
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document.write( "We can split the area into two triangles.\r<BR>\n" );
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document.write( "* Triangle 1: Vertices $(-\frac{7}{3}, \frac{7}{3})$, $(-\frac{1}{2}, 6)$, and $(0, 0)$.<BR>\n" );
document.write( "* Triangle 2: Vertices $(0, 0)$, $(-\frac{1}{2}, 6)$, and $(\frac{5}{3}, \frac{5}{3})$.\r<BR>\n" );
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document.write( "Area of Triangle 1:<BR>\n" );
document.write( "Using the determinant formula:<BR>\n" );
document.write( "$$ \frac{1}{2} \left| (-\frac{7}{3})(6 - 0) + (-\frac{1}{2})(0 - \frac{7}{3}) + 0(\frac{7}{3} - 6) \right| $$<BR>\n" );
document.write( "$$ \frac{1}{2} \left| -14 + \frac{7}{6} \right| = \frac{1}{2} \left| \frac{-84 + 7}{6} \right| = \frac{1}{2} \left| \frac{-77}{6} \right| = \frac{77}{12} $$\r<BR>\n" );
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document.write( "Area of Triangle 2:<BR>\n" );
document.write( "$$ \frac{1}{2} \left| 0(6 - \frac{5}{3}) + (-\frac{1}{2})(\frac{5}{3} - 0) + (\frac{5}{3})(0 - 6) \right| $$<BR>\n" );
document.write( "$$ \frac{1}{2} \left| -\frac{5}{6} - 10 \right| = \frac{1}{2} \left| \frac{-5 - 60}{6} \right| = \frac{1}{2} \left| \frac{-65}{6} \right| = \frac{65}{12} $$\r<BR>\n" );
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document.write( "Total area:<BR>\n" );
document.write( "$$ \frac{77}{12} + \frac{65}{12} = \frac{142}{12} = \frac{71}{6} $$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\frac{71}{6}}$<BR>\n" );
document.write( " </SPAN>\n" );
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