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Question 1109244: https://imgur.com/a/H4UUq
The original statement was an OR statement, which lead me to conclude then that I inverted/negated ALL statements. Namely,
x>= -1 AND x<1
as opposed to
x<=-1 OR x>1
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?
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. :(
Why do the 2nd and 3rd answers not include the word AND? Is there a significance?
I thought that to negate an equality sign, you merely flipped it. > becomes <, and <= becomes >=.
The correct answer was the 2nd answer, which my reading of it means that:
"x is less than -1 AND x is less than or equal to 1."
Going with my (flawed I assume) logic above of merely flipping the equality operators I thought this would have actually have been
x is less than or equal to -1 AND X is greater than 1"
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
demorgan's law states:
not (A OR B) = not (A) AND not (B).
not(A AND B) = not (A) OR not (B).
your original statement is:
(x <= -1 OR x > 1)
the demorgan law that applies is:
not (A OR B) = not (A) AND not (B).
if you let A = (x <= -1) and B = (x > 1), then you get:
not((x <= -1) OR (x > 1)) = not (x <= -1) AND not (x > 1).
not(x <= -1) becomes (x > -1)
not(x > 1) becomes (x <= 1)
your equation of:
not((x <= -1) OR (x > 1)) = not (x <= -1) AND not (x > 1) becomes:
not((x <= -1) OR (x > 1)) = (x > -1) AND (x <= 1).
x > -1 is the same as -1 < x.
therefore (x > -1) AND (x <= 1) becomes:
(-1 < x) AND (x <= 1).
this can be written as -1 < x <= 1.
that would be the same as the second selection.
your questions are:
question 1:
Why do the 2nd and 3rd answers not include the word AND? Is there a significance?
the 2d answer is -1 < x <= 1
that is the same as saying that -1 < x AND x <= 1.
the 3d answer is -1 <= x < 1.
that is the same as saying that -1 <= x AND x < 1.
the AND, in either one of these cases, is unnecessary.
question 2:
I thought that to negate an equality sign, you merely flipped it. > becomes <, and <= becomes >=.
you are probably thinking of what happens when you multiply both sides of an inequality by a negative number.
if your inequality is x < y and you multiply both sides by -1, then you get -x > -y.
if your inequality is x <= y and you multiply both sides by -1, then you get -x >= -y
in both cases, you are reversing the inequality.
that is not the same as negating the inequality itself.
a negation of x < y means x >= y.
a negation of x <= y means x > y.
a negation of x > y means x <= y.
a negation of x >= y means x < y.
if x is smaller than y, then x cannot be equal to y nor can it be greater than y.
therefore, the negation of x smaller than y means that x can either be greater than y or equal to y.
if x is greater than y or equal to y, x cannot be smaller than y.
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