document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=554427>Question 554427</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to solve the inequality -4(2x+7)<16? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #361353 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=361353\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given to solve:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A%282%2Ax%2B7%29%3C16 ALIGN=MIDDLE  ALT=\"-4%2A%282%2Ax%2B7%29%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "In general, you can work inequalities such as this one using the same rules as you would for equations with the exception that if you divide or multiply both sides by a negative quantity, the sign of the inequality will reverse direction. <BR>\n" );
document.write( ".<BR>\n" );
document.write( "So, let's begin by doing the distributed multiplication on the left side. Multiplying both terms inside the parentheses by -4 results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8%2Ax+-+28+%3C+16 ALIGN=MIDDLE  ALT=\"-8%2Ax+-+28+%3C+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next, let's get rid of the -28 on the left side by adding +28 to both sides. The -28 and the +28 will cancel each other on the left side and on the right side the +16 and the +28 total to +44. So the inequality is now:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8%2Ax+%3C+44 ALIGN=MIDDLE  ALT=\"-8%2Ax+%3C+44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Just as we would do for an equation, we can solve for x by dividing both sides by -8. But remember the rule that if we divide or multiply both sides by a minus quantity, the direction of the inequality sign changes. <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Dividing the -8x by -8 results in just x and dividing 44 by -8 gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. And reversing the direction of the inequality sign due to the division by the MINUS 8 makes it change direction so it points to the right as shown:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+-5%261%2F2 ALIGN=MIDDLE  ALT=\"x+%3E+-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "This tells us that the original inequality that we were given will only be true for values of x that are greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. On the number line, such values lie to the right of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's raise our confidence in that answer by taking the original inequality and substituting a value for x that is greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to see if it works. Let's try letting x equal -5:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Start with:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A%282%2Ax%2B7%29%3C16 ALIGN=MIDDLE  ALT=\"-4%2A%282%2Ax%2B7%29%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Replace x by -5 to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A%282%2A%28-5%29%2B7%29%3C16 ALIGN=MIDDLE  ALT=\"-4%2A%282%2A%28-5%29%2B7%29%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Inside the parentheses the 2 multiplies the -5 to give -10 and the +7 adds to the -10 to result in -3. This leads to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A%28-10%2B7%29%3C16 ALIGN=MIDDLE  ALT=\"-4%2A%28-10%2B7%29%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and then:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A%28-3%29%3C16 ALIGN=MIDDLE  ALT=\"-4%2A%28-3%29%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Multiplying the -4 times the -3 gives +12 so when x equals -5, the inequality becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%3C16 ALIGN=MIDDLE  ALT=\"12%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That certainly is true ... 12 is less than 16. So for a value of x that is slightly greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > the original inequality that we were given is true.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "For practice, why don't you see what happens when x is replaced by something slightly less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > meaning that it lies slightly to the left of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the number line. If you try letting x equal -6 in the original inequality, you should get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20+%3C+16 ALIGN=MIDDLE  ALT=\"20+%3C+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "But that is not true. +20 is greater than +16. So x cannot be -6 and still have the original inequality be true. Somewhere between x equals -5 and x equals -6, the value of x will make the inequality go from being true to being untrue. That helps to give us confidence that our answer is correct. x must be greater than some value between -5 and -6. <BR>\n" );
document.write( ".<BR>\n" );
document.write( "For more practice, why don't you try letting x be exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and see what happens when you substitute that value for x into the original inequality? You should get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%3C16 ALIGN=MIDDLE  ALT=\"16%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That is not true. +16 is equal to, not less than +16. This tells you that, on the number line you must go a little to the right of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%261%2F2 ALIGN=MIDDLE  ALT=\"-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to make the original inequality be true. This establishes that our answer is correct. Namely, it must be that:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E-5%261%2F2 ALIGN=MIDDLE  ALT=\"x+%3E-5%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to understand the problem a little better. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );