document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1109244>Question 1109244</A>: <I><SPAN STYLE=\"word-break: break-all;\"> https://imgur.com/a/H4UUq\r<BR>\n" );
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document.write( "The original statement was an OR statement, which lead me to conclude then that I inverted/negated ALL statements. Namely,\r<BR>\n" );
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document.write( "x>= -1 AND x<1\r<BR>\n" );
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document.write( "as opposed to\r<BR>\n" );
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document.write( "x<=-1 OR x>1\r<BR>\n" );
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document.write( "And that it was on that basis that I selected the final answer (which has been identified as being wrong). Where did I go wrong?\r<BR>\n" );
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document.write( "I then thought perhaps the answer was the 1st one by virtue of the fact that it also had an \"and\" statement contained within it, but I must confess to being baffled now. :(\r<BR>\n" );
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document.write( "Why do the 2nd and 3rd answers not include the word AND? Is there a significance?\r<BR>\n" );
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document.write( "I thought that to negate an equality sign, you merely flipped it. > becomes <, and <= becomes >=.\r<BR>\n" );
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document.write( "The correct answer was the 2nd answer, which my reading of it means that:\r<BR>\n" );
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document.write( "\"x is less than -1 AND x is less than or equal to 1.\"\r<BR>\n" );
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document.write( "Going with my (flawed I assume) logic above of merely flipping the equality operators I thought this would have actually have been\r<BR>\n" );
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document.write( "x is less than or equal to -1 AND X is greater than 1\"\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #724246 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=724246\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "demorgan's law states:\r<BR>\n" );
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document.write( "not (A OR B) = not (A) AND not (B).\r<BR>\n" );
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document.write( "not(A AND B) = not (A) OR not (B).\r<BR>\n" );
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document.write( "your original statement is:\r<BR>\n" );
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document.write( "(x <= -1 OR x > 1)\r<BR>\n" );
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document.write( "the demorgan law that applies is:\r<BR>\n" );
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document.write( "not (A OR B) = not (A) AND not (B).\r<BR>\n" );
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document.write( "if you let A = (x <= -1) and B = (x > 1), then you get:\r<BR>\n" );
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document.write( "not((x <= -1) OR (x > 1)) = not (x <= -1) AND not (x > 1).\r<BR>\n" );
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document.write( "not(x <= -1) becomes (x > -1)\r<BR>\n" );
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document.write( "not(x > 1) becomes (x <= 1)\r<BR>\n" );
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document.write( "your equation of:\r<BR>\n" );
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document.write( "not((x <= -1) OR (x > 1)) = not (x <= -1) AND not (x > 1) becomes:\r<BR>\n" );
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document.write( "not((x <= -1) OR (x > 1)) = (x > -1) AND (x <= 1).\r<BR>\n" );
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document.write( "x > -1 is the same as -1 < x.\r<BR>\n" );
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document.write( "therefore (x > -1) AND (x <= 1) becomes:\r<BR>\n" );
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document.write( "(-1 < x) AND (x <= 1).\r<BR>\n" );
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document.write( "this can be written as -1 < x <= 1.\r<BR>\n" );
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document.write( "that would be the same as the second selection.\r<BR>\n" );
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document.write( "your questions are:\r<BR>\n" );
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document.write( "question 1:\r<BR>\n" );
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document.write( "Why do the 2nd and 3rd answers not include the word AND? Is there a significance? \r<BR>\n" );
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document.write( "the 2d answer is -1 < x <= 1\r<BR>\n" );
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document.write( "that is the same as saying that -1 < x AND x <= 1.\r<BR>\n" );
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document.write( "the 3d answer is -1 <= x < 1.\r<BR>\n" );
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document.write( "that is the same as saying that -1 <= x AND x < 1.\r<BR>\n" );
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document.write( "the AND, in either one of these cases, is unnecessary.\r<BR>\n" );
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document.write( "question 2:\r<BR>\n" );
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document.write( "I thought that to negate an equality sign, you merely flipped it. > becomes <, and <= becomes >=. \r<BR>\n" );
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document.write( "you are probably thinking of what happens when you multiply both sides of an inequality by a negative number.\r<BR>\n" );
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document.write( "if your inequality is x < y and you multiply both sides by -1, then you get -x > -y.\r<BR>\n" );
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document.write( "if your inequality is x <= y and you multiply both sides by -1, then you get -x >= -y\r<BR>\n" );
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document.write( "in both cases, you are reversing the inequality.\r<BR>\n" );
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document.write( "that is not the same as negating the inequality itself.\r<BR>\n" );
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document.write( "a negation of x < y means x >= y.\r<BR>\n" );
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document.write( "a negation of x <= y means x > y.\r<BR>\n" );
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document.write( "a negation of x > y means x <= y.\r<BR>\n" );
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document.write( "a negation of x >= y means x < y.\r<BR>\n" );
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document.write( "if x is smaller than y, then x cannot be equal to y nor can it be greater than y.\r<BR>\n" );
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document.write( "therefore, the negation of x smaller than y means that x can either be greater than y or equal to y.\r<BR>\n" );
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document.write( "if x is greater than y or equal to y, x cannot be smaller than y.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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