Proof: Inequality Problem 3
Let's prove the following theorem:
if y ⋅ (-2) < -16, then y > 8
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:
Given
| 1 | y ⋅ (-2) < -16 |
|---|
| # | Claim | Reason |
|---|---|---|
| 1 | (-1) / 2 < 0 | (-1) / 2 < 0 |
| 2 | (y ⋅ (-2)) ⋅ ((-1) / 2) > (-16) ⋅ ((-1) / 2) | if y ⋅ (-2) < -16 and (-1) / 2 < 0, then (y ⋅ (-2)) ⋅ ((-1) / 2) > (-16) ⋅ ((-1) / 2) |
| 3 | (-2) ⋅ ((-1) / 2) = 1 | (-2) ⋅ ((-1) / 2) = 1 |
| 4 | (-16) ⋅ ((-1) / 2) = 8 | (-16) ⋅ ((-1) / 2) = 8 |
| 5 | (y ⋅ (-2)) ⋅ ((-1) / 2) = y ⋅ 1 | if (-2) ⋅ ((-1) / 2) = 1, then (y ⋅ (-2)) ⋅ ((-1) / 2) = y ⋅ 1 |
| 6 | (y ⋅ (-2)) ⋅ ((-1) / 2) = y | if (y ⋅ (-2)) ⋅ ((-1) / 2) = y ⋅ 1, then (y ⋅ (-2)) ⋅ ((-1) / 2) = y |
| 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 |
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