To begin with, let us state an important property of inequalities: inequalities are transitive. This means that, if a > b and if b > c\ then a > c. Since equalities (equations) are also transitive, the transitivity property should apply to "weak" inequalities (> or <) as well as to "strict" ones (> or <)t Thus have a > h, b > c => a > c a >b,h >c a >c
This property is what makes possible the writing of a continued inequality, such as 3 < a < b < 8 or 7 < x < 24. (In writing a continued inequality, the inequality signs are as a rule arranged in the same direction, usually with the smallest number on the left.)
The most important rules of inequalities are those governing the addition (subtraction) of a number to (from) an inequality, the multiplication or division of an inequality by a number, and the squaring of an inequality. Specifically; these rules are as follows.
An inequality will continue to hold if an equal quantity is added to or subtracted from each side. This rule may be generalized thus: \(a > b > c\ then a±k>b±k>c±k.
Rule il (multiplication and division)
I ka> kh (k > 0) * > 1 ka < kh (k < 0)
The multiplication of both sides by a positive number preserves the inequality, but a negative multiplier will cause the sense (or direction) of the inequality to be reversed.
Example 1 ^nce ® > 5' multiplication by 3 will yield 3(6) > 3(5), or 18 > 15; but multiplication by -3 -"- will result in (—3)6 < (-3)5, or -18 < -15.
Division of an inequality by a number n is equivalent to multiplication by the number 1 /n; therefore ihe rule on division is subsumed under the rule on multiplication.
Rule III (squaring)
If its two sides are both nonnegativc, the inequality will continue to hold when both sides are squared.
Since 4 > 3 and since both sides are positive, we have A2 > 32, or 16 > 9. Similarly, since 2 > 0, it follows that 21 > G\ or 4 > 0.
Rules 1 through III have been stated in terms of strict inequalities, but their validity is unaffected if the > signs are replaced by > signs,
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