Proof: Inequality Problem 3

Let's prove the following theorem:

The inequality properties allow us to multiply -1/2 to both sides, but we need to flip the operator to the greater-than sign (>).

y ⋅ -2 ⋅ -1/2 = y

and

-16 ⋅ -1/2 = 8

so

y > 8

Proof:

Proof Table

#	Claim	Reason
1	(-1) / 2 < 0	(-1) / 2 < 0 (Fixed Value Statement)
2	(y ⋅ (-2)) ⋅ ((-1) / 2) > (-16) ⋅ ((-1) / 2)	if y ⋅ (-2) < -16 and (-1) / 2 < 0, then (y ⋅ (-2)) ⋅ ((-1) / 2) > (-16) ⋅ ((-1) / 2) (Inequality Multiplication By Negative Value)
3	(-2) ⋅ ((-1) / 2) = 1	(-2) ⋅ ((-1) / 2) = 1 (Fixed Value Statement)
4	(-16) ⋅ ((-1) / 2) = 8	(-16) ⋅ ((-1) / 2) = 8 (Fixed Value Statement)
5	(y ⋅ (-2)) ⋅ ((-1) / 2) = y ⋅ 1	if (-2) ⋅ ((-1) / 2) = 1, then (y ⋅ (-2)) ⋅ ((-1) / 2) = y ⋅ 1 (Apply Associative Multiply)
6	(y ⋅ (-2)) ⋅ ((-1) / 2) = y	if (y ⋅ (-2)) ⋅ ((-1) / 2) = y ⋅ 1, then (y ⋅ (-2)) ⋅ ((-1) / 2) = y (Multiply by 1)
7	y > 8	if (y ⋅ (-2)) ⋅ ((-1) / 2) > (-16) ⋅ ((-1) / 2) and (y ⋅ (-2)) ⋅ ((-1) / 2) = y and (-16) ⋅ ((-1) / 2) = 8, then y > 8 (Double Inequality 2)

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