document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209903>Question 1209903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of points of intersection between the graphs of the following equations: \begin{align*}<BR>\n" );
document.write( "y = |2x + 5|,<BR>\n" );
document.write( "y = -|3x - 2| + |4x - 7| + x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851009 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=851009\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the given equations to find the number of intersection points.\r<BR>\n" );
document.write( "\n" );
document.write( "**Equation 1:** $y = |2x + 5|$\r<BR>\n" );
document.write( "\n" );
document.write( "**Equation 2:** $y = -|3x - 2| + |4x - 7| + x$\r<BR>\n" );
document.write( "\n" );
document.write( "We need to consider cases based on the values of $x$ that make the expressions inside the absolute values zero.\r<BR>\n" );
document.write( "\n" );
document.write( "**Critical Points:**\r<BR>\n" );
document.write( "\n" );
document.write( "* $2x + 5 = 0 \Rightarrow x = -5/2 = -2.5$<BR>\n" );
document.write( "* $3x - 2 = 0 \Rightarrow x = 2/3$<BR>\n" );
document.write( "* $4x - 7 = 0 \Rightarrow x = 7/4 = 1.75$\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 1: $x \le -5/2$**\r<BR>\n" );
document.write( "\n" );
document.write( "* $|2x + 5| = -(2x + 5) = -2x - 5$<BR>\n" );
document.write( "* $|3x - 2| = -(3x - 2) = -3x + 2$<BR>\n" );
document.write( "* $|4x - 7| = -(4x - 7) = -4x + 7$\r<BR>\n" );
document.write( "\n" );
document.write( "Equation 1: $y = -2x - 5$<BR>\n" );
document.write( "Equation 2: $y = -(-3x + 2) + (-4x + 7) + x = 3x - 2 - 4x + 7 + x = 5$\r<BR>\n" );
document.write( "\n" );
document.write( "Set the equations equal:<BR>\n" );
document.write( "$-2x - 5 = 5$<BR>\n" );
document.write( "$-2x = 10$<BR>\n" );
document.write( "$x = -5$\r<BR>\n" );
document.write( "\n" );
document.write( "Since $-5 \le -2.5$, this is a valid solution.<BR>\n" );
document.write( "Intersection point: $(-5, 5)$\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 2: $-5/2 < x \le 2/3$**\r<BR>\n" );
document.write( "\n" );
document.write( "* $|2x + 5| = 2x + 5$<BR>\n" );
document.write( "* $|3x - 2| = -(3x - 2) = -3x + 2$<BR>\n" );
document.write( "* $|4x - 7| = -(4x - 7) = -4x + 7$\r<BR>\n" );
document.write( "\n" );
document.write( "Equation 1: $y = 2x + 5$<BR>\n" );
document.write( "Equation 2: $y = -(-3x + 2) + (-4x + 7) + x = 3x - 2 - 4x + 7 + x = 5$\r<BR>\n" );
document.write( "\n" );
document.write( "Set the equations equal:<BR>\n" );
document.write( "$2x + 5 = 5$<BR>\n" );
document.write( "$2x = 0$<BR>\n" );
document.write( "$x = 0$\r<BR>\n" );
document.write( "\n" );
document.write( "Since $-2.5 < 0 \le 2/3$, this is a valid solution.<BR>\n" );
document.write( "Intersection point: $(0, 5)$\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 3: $2/3 < x \le 7/4$**\r<BR>\n" );
document.write( "\n" );
document.write( "* $|2x + 5| = 2x + 5$<BR>\n" );
document.write( "* $|3x - 2| = 3x - 2$<BR>\n" );
document.write( "* $|4x - 7| = -(4x - 7) = -4x + 7$\r<BR>\n" );
document.write( "\n" );
document.write( "Equation 1: $y = 2x + 5$<BR>\n" );
document.write( "Equation 2: $y = -(3x - 2) + (-4x + 7) + x = -3x + 2 - 4x + 7 + x = -6x + 9$\r<BR>\n" );
document.write( "\n" );
document.write( "Set the equations equal:<BR>\n" );
document.write( "$2x + 5 = -6x + 9$<BR>\n" );
document.write( "$8x = 4$<BR>\n" );
document.write( "$x = 1/2 = 0.5$\r<BR>\n" );
document.write( "\n" );
document.write( "However, $2/3 \approx 0.666$, so $0.5$ is not in the interval, so no solution in this case.\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 4: $x > 7/4$**\r<BR>\n" );
document.write( "\n" );
document.write( "* $|2x + 5| = 2x + 5$<BR>\n" );
document.write( "* $|3x - 2| = 3x - 2$<BR>\n" );
document.write( "* $|4x - 7| = 4x - 7$\r<BR>\n" );
document.write( "\n" );
document.write( "Equation 1: $y = 2x + 5$<BR>\n" );
document.write( "Equation 2: $y = -(3x - 2) + (4x - 7) + x = -3x + 2 + 4x - 7 + x = 2x - 5$\r<BR>\n" );
document.write( "\n" );
document.write( "Set the equations equal:<BR>\n" );
document.write( "$2x + 5 = 2x - 5$<BR>\n" );
document.write( "$5 = -5$\r<BR>\n" );
document.write( "\n" );
document.write( "This is a contradiction, so there are no solutions in this case.\r<BR>\n" );
document.write( "\n" );
document.write( "**Intersection Points:**\r<BR>\n" );
document.write( "\n" );
document.write( "* $(-5, 5)$<BR>\n" );
document.write( "* $(0, 5)$\r<BR>\n" );
document.write( "\n" );
document.write( "There are two intersection points.\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{2}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );