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1. (a) 2x - 4 = 2 x = 3 (b) x = 3 2x - 4 = 2 (c) x = 1 => x2 - 2x + 1 = 0 (d) x2 > 4 <=> x > 2 or x < -2

3. (a) x > 0 is necessary, but not sufficient, (b) x > 50 is sufficient, but not necessary, (c) x > 4 is necessary and sufficient.

5. (a) x = 3 (b) x = 0 or x = -5 (c) x = -3 or x = 3

7. (a) Iff. (Note: Ja means 2, not ±2.) (b) If (c) Only if (d) Iff (e) Only if

9. (a) x < 0 or y <0 (b) jc < a for at least one x. (c) x and/or y is less than 5. (Is it easier if the statement were: neither Ron nor Rita is less than 5 years old?) (d) There exists an s > 0 such that B is not satisfied for any 8 > 0. (e) Someone may not like cats, (f) Someone never loves anyone.

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