Determine whether the given pair of statements are equivalent
(a) If |x| > 1, then x > 1 or x < −1.
If −1 ≤ x ≤ 1, then |x| ≤ 1.
(b) If 2x + 5 = 15, then x = 5.
If 2x + 5 ≠ 15, then x ≠ 5.
(a) If |x| > 1, then x > 1 or x < −1.
If −1 ≤ x ≤ 1, then |x| ≤ 1.
(b) If 2x + 5 = 15, then x = 5.
If 2x + 5 ≠ 15, then x ≠ 5.
Answers & Comments
Answer:
(a) If |x| > 1, then x > 1 or x < −1.
If −1 ≤ x ≤ 1, then |x| ≤ 1.
These two statements are equivalent. The first statement says that if the absolute value of x is greater than 1, then x is either greater than 1 or less than -1. The second statement says that if x is between -1 and 1 (inclusive), then the absolute value of x is less than or equal to 1. These two statements say the same thing in different ways.
(b) If 2x + 5 = 15, then x = 5.
If 2x + 5 ≠ 15, then x ≠ 5.
These two statements are also equivalent. The first statement says that if 2x + 5 equals 15, then x must be 5. The second statement says that if 2x + 5 does not equal 15, then x cannot be 5. These two statements are just different ways of saying the same thing.