If p and q locate points on a horizontal line and p is less than q, p < q , then we consider p on the left of q.
Definition 8. p is less than q written p < q ⇔ there is a positive number r such that p + r = q.
Definition 9. p is greater than q written p > q ⇔ there is a positive number r such that p = q + r .
Definition 10. p is less than or equal to q written p ≤ q ⇔ p = q or p < q
Definition 11. p is greater than or equal to q written p ≥ q ⇔ p = q or p > q
Definition 11. p is NOT (an INEQUALITY above) q written p (that INEQUALITY with a vertical line through it) q
⇔ a different INEQUALITY above that you can determine correct by negation.
LVI. p is less than 0 ⇔ p < 0 ⇔ p is negative.
LII. RESTATED: | p | ≥ 0
LVII. | p | = r ⇔ p = r or p = -r where r ≥ 0
XVII. RESTATED: If p < q and q < r, then p < r.
LVIII. If p < q and r < t, then p + r < q + t.
* may replace < with any other inequality and + with - within reason.
LIX. If p < q and r > 0, then p r < q r. ( see * above for general idea )
LX. If p < q and r < 0, then p r > q r. ( see * above for general idea )
* Properties XVII, LVIII, LIX and LX are prototypes for many other inequality properties.
LXI. | p - q | = r ⇔ p = q - r or p = q + r
LXII. | p - q | < r ⇔ q - r < p < q + r
LXIII. | p - q | > r ⇔ p < q - r or p > q + r
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