document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=217047>Question 217047</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What integers are solutions to |a - 12| < 9 AND |a + 2| > 17? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #163755 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=163755\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Replace a with x and your equations to solve are:<BR>\n" );
document.write( "|x-12| < 9 AND |x+2| > 17<BR>\n" );
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document.write( "I do this because I'm so used to working with x that I always write x even when I know it's a or b or whatever.<BR>\n" );
document.write( "Converting to x eliminates the problems IF I remember to convert back at the end.<BR>\n" );
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document.write( "Pure laziness on my part because I already started using x and didn't want to have to go back and convert them to a in all the equations.<BR>\n" );
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document.write( "The operating principle is:<BR>\n" );
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document.write( "The absolute value of an expression equals the expression if the expression is greater than or equal to 0.<BR>\n" );
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document.write( "The absolute value of an expression equals minus the expression if the expression is less than or equal to 0.<BR>\n" );
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document.write( "let E be the expression.<BR>\n" );
document.write( "Let the equation be |E| > A<BR>\n" );
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document.write( "If E >= 0, we remove the absolute value sign and get E > A.<BR>\n" );
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document.write( "If E < 0, we remove the absolute value sign and set -E > A.<BR>\n" );
document.write( "We multiply both sides of this equation by (-1) to get E < -A.<BR>\n" );
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document.write( "We need to find when the expressions within the absolute value signs are positive and when the expressions within the absolute value signs are negative.<BR>\n" );
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document.write( "You have a maximum of 4 possible cases.<BR>\n" );
document.write( "case 1 is when both expressions are positive within the absolute value signs.<BR>\n" );
document.write( "case 2 is when both expressions are negative within the absolute value signs.<BR>\n" );
document.write( "case 3 is when the first expression is positive within the absolute value signs, and the second expression is negative within the absolute value signs.<BR>\n" );
document.write( "case 4 is when the first expression is negative within the absolute value signs, and the second expression is positive within the absolute value signs.<BR>\n" );
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document.write( "We test each expression separately to find out when the expression within the absolute value signs is positive and when the expression within the absolute value signs is negative.<BR>\n" );
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document.write( "Take |x-12| < 9<BR>\n" );
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document.write( "The expression within the absolute value sign is (x-12).  This is our first expression.<BR>\n" );
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document.write( "x-12 >= 0 when x >= 12<BR>\n" );
document.write( "x-12 < 0 when x < 12<BR>\n" );
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document.write( "The first expression is positive when x >= 12<BR>\n" );
document.write( "The first expression is negative when x < 12<BR>\n" );
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document.write( "Now take |x+2| > 17<BR>\n" );
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document.write( "The expression within the absolute value sign is (x+2). This is our second expression.<BR>\n" );
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document.write( "x+2 >= 0 when x >= -2<BR>\n" );
document.write( "x+2 < 0 when x < -2<BR>\n" );
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document.write( "The second expression is positive when x >= -2.<BR>\n" );
document.write( "The second expression is negative when x < -2<BR>\n" );
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document.write( "Our 4 cases reduce to 3.<BR>\n" );
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document.write( "They are:<BR>\n" );
document.write( "Case 1:<BR>\n" );
document.write( "Both expressions are negative when x < -2.<BR>\n" );
document.write( "Case 2:<BR>\n" );
document.write( "Both expressions are positive when x >= 12.<BR>\n" );
document.write( "Case 3:<BR>\n" );
document.write( "The first expression is negative when x < 12 and the second expression is positive when x >= -2<BR>\n" );
document.write( "Case 4:<BR>\n" );
document.write( "We do not have a case when first expression is positive and second expression is negative so this one cancels out.<BR>\n" );
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document.write( "We look at each of these 3 cases separately.<BR>\n" );
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document.write( "case 1 (both expressions are positive within the absolute value signs).<BR>\n" );
document.write( "This happens when x >= 12.<BR>\n" );
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document.write( "|x-12| < 9 becomes (x-12) < 9 which becomes x < 21.<BR>\n" );
document.write( "|x+2| > 17 becoming (x+2) > 17 which becomes x > 15.<BR>\n" );
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document.write( "We have 2 possible solutions.<BR>\n" );
document.write( "They are:<BR>\n" );
document.write( "x < 21 or x > 15.<BR>\n" );
document.write( "Since case 1 is when x >= 12, both these solutions are possible.<BR>\n" );
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document.write( "We'll move to case number 2 which is when both expressions are negative within the absolute value signs.<BR>\n" );
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document.write( "Case 2 (both expressions are negative within the absolute value signs).<BR>\n" );
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document.write( "This happens when x < -2<BR>\n" );
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document.write( "|x-12| < 9 becomes -(x-12) < 9 becomes (x-12) > -9 when you multiply both sides of this equation by (-1).  This becomes x > 3<BR>\n" );
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document.write( "|x+2| > 17 becomes -(x+2) > 17 which becomes (x+2) < -17 when you multiply both sides of this equation by (-1).  This becomes x < -19<BR>\n" );
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document.write( "Our two possible solutions are:<BR>\n" );
document.write( "x > 3 or x < -19<BR>\n" );
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document.write( "Since case 2 is when x < -2, the only possible solution is x < -19.<BR>\n" );
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document.write( "We'll move on to case 3 and see what we get.<BR>\n" );
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document.write( "Case 3 is: (x-12) is negative and (x+2) is positive.<BR>\n" );
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document.write( "This happens when x >= 2 and x < 12<BR>\n" );
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document.write( "Since (x-12) is negative within the absolute value signs, we get:<BR>\n" );
document.write( "|x-12| < 9 becomes -(x-12) < 9 becomes (x-12) > -9 becomes x > 3.<BR>\n" );
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document.write( "Since (x+2) > 17 is positive within the absolute value signs, we get:<BR>\n" );
document.write( "|x+2| > 17 becomes (x+2) > 17 becomes x > 15.<BR>\n" );
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document.write( "We have two possible solutions of x > 15 and x > 3<BR>\n" );
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document.write( "Since Case 2 is when x >= 2 ANDE x < 12, the only possible solution is x > 3 because x > 15 is outside the range of permissible values for Case 2.<BR>\n" );
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document.write( "We have 3 possible solutions for this set of equations.<BR>\n" );
document.write( "They are:<BR>\n" );
document.write( "x > 15 and x < 21<BR>\n" );
document.write( "x < -19<BR>\n" );
document.write( "x > 3<BR>\n" );
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document.write( "We look at each in turn.<BR>\n" );
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document.write( "Our equations are:<BR>\n" );
document.write( "|x-12| < 9<BR>\n" );
document.write( "|x+2| > 17<BR>\n" );
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document.write( "When x = -20 < -19, we get:<BR>\n" );
document.write( "|x-12| = |-20-12| = |-32| = 32 NOT < 9.<BR>\n" );
document.write( "This solution is not good because it does not satisfy one of the equations.<BR>\n" );
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document.write( "When x = 4 > 3, we get:<BR>\n" );
document.write( "|x-12| = |4-12| = |-8| = 8 < 9 which is OK so we check the second equation.<BR>\n" );
document.write( "|x+2| = |4+2| = |6| = 6 NOT > 17.<BR>\n" );
document.write( "This solution is not good because it does not satisfy one of the equations.<BR>\n" );
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document.write( "When x = 16 > 15 and < 21, we get:<BR>\n" );
document.write( "|x-12| = |16-12| = |4| = 4< 9 OK.<BR>\n" );
document.write( "|x+2| = |16+2| = |18| = 18 > 17 OK.<BR>\n" );
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document.write( "So far, when x > 15 and < 21, we can solve both equations simultaneously.<BR>\n" );
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document.write( "We tested the lower limit of 16 for this possible solution set, so we test  the upper limit of these solutions by testing when x = 20.<BR>\n" );
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document.write( "When x = 20 > 15 < 21, we get:<BR>\n" );
document.write( "|x-12| = |20-12| = 8 < 9 which is good.<BR>\n" );
document.write( "|x+2| = |20+2| = 22 > 17 which is also good.<BR>\n" );
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document.write( "Within the limits we are ok.<BR>\n" );
document.write( "We test outside the limits next to see if the limits were expressed correctly.<BR>\n" );
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document.write( "When x = 21, we get:<BR>\n" );
document.write( "|x-12| = 9 NOT < 9 which is not good.  The limits on the high end are correct.<BR>\n" );
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document.write( "When x = 15, we get:<BR>\n" );
document.write( "|x-12| = 3 < 9 which is good.<BR>\n" );
document.write( "|x+2| > 17 = 17 NOT > 17 which is not good.  The limits on the low end are correct.<BR>\n" );
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document.write( "Before we answer the question, we set x = a to convert back to the original variable name.<BR>\n" );
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document.write( "Our answer to the question is:<BR>\n" );
document.write( "15 < a < 21<BR>\n" );
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document.write( "Those values of a will satisfy both equations of:<BR>\n" );
document.write( "|a-12| < 9<BR>\n" );
document.write( "and<BR>\n" );
document.write( "|a+2| > 17<BR>\n" );
document.write( "simultaneously.<BR>\n" );
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document.write( " </SPAN>\n" );
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