Consider the statement "Either −2≤ 𝑥 ≤−1 or 1 ≤ 𝑥 ≤2" The negation of this…
2016
Consider the statement
"Either −2≤ 𝑥 ≤−1 or 1 ≤ 𝑥 ≤2"
The negation of this statement is
- A.
x < -2 or 2 < x or -1 < x <1
- B.
x < -2 or 2 < x
- C.
-1 < x < 1
- D.
x ≤ -2 or 2 ≤ x or -1 < x <1
Attempted by 186 students.
Show answer & explanation
Correct answer: A
Negate the statement using De Morgan's law:
Write the original statement as A or B where A is "-2 ≤ x ≤ -1" and B is "1 ≤ x ≤ 2".
Negate: not(A or B) = (not A) and (not B) by De Morgan's law.
Find not A: the negation of -2 ≤ x ≤ -1 is x < -2 or x > -1.
Find not B: the negation of 1 ≤ x ≤ 2 is x < 1 or x > 2.
Intersect (not A) and (not B): (x < -2 or x > -1) and (x < 1 or x > 2).
Combine the cases to get the complement of the two closed intervals: x < -2, or -1 < x < 1, or x > 2.
Final answer: x < -2 or -1 < x < 1 or x > 2. (Endpoints -2, -1, 1, 2 are excluded because they were included in the original statement.)
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