State, whether the following statements are true or false.

(i) If a < b, then a - c < b - c

(ii) If a > b, then a + c > b + c

(iii) If a < b, then ac > bc

(iv) If a > b, then $\dfrac{a}{c} \lt \dfrac{b}{c}$

(v) If a - c > b - d; then a + d > b + c

(vi) If a < b, and c > 0, then a - c > b - c

where a, b, c, and d are real numbers c ≠ 0.

(i) Given,

a < b

Subtracting both sides by c,

a - c < b - c.

Hence, the statement is True.

(ii) Given,

a > b

Adding both sides by c,

a + c > b + c.

Hence, the statement is True.

(iii) Given,

a < b

If c is a positive number,

Multiplying both sides by c we get,

ac < bc

If c is a negative number,

Multiplying both sides by c we get,

ac > bc [Using rule 4]

Hence, the statement is False.

(iv) Given,

a > b

If c is a positive number,

Dividing both sides by c we get,

$\dfrac{a}{c} \gt \dfrac{b}{c}$

If c is a negative number,

Dividing both sides by c we get,

$\dfrac{a}{c} \lt \dfrac{b}{c}$ [Using rule 4]

Hence, the statement is False.

(v) Given,

a - c > b - d

Adding both sides by (c + d) we get,

⇒ a - c + (c + d) > b - d + (c + d)

⇒ a - c + c + d > b + c - d + d

⇒ a + d > b + c

Hence, the statement is True.

(vi) Given,

a < b and c > 0

Subtracting both sides by c we get,

a - c < b - c [As c is a positive number.]

Hence, the statement is False.