Inequality Morphing

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Some of the familiar equation-morphing rules also work for inequalities, but others must be modified, and a few don t work at all Here are the things we can do with two-part equations, summarized for reference Reverse the order Add the same quantity to both sides Subtract the same quantity from both sides Add one equation to another Multiply both sides by the same quantity Divide both sides by the same nonzero quantity

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Whenever we do one or more of these things to an equation, we get another valid equation Now let s see how well these rules work for inequalities (We won t get into formal proofs of these facts) You can try out some examples if you want to improve your understanding of how they work

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Manipulating statements If two quantities a and b are different, we can express that fact in either order In general, it is always true that if a b, then b a We can add or subtract the same quantity from each side of a not equal statement If two quantities are different to start out with, then they ll still be different if we add or subtract the same quantity from both If a b, then for any number c

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a+c b+c and a c b c We cannot, in general, add two not equal statements and get another not equal statement Consider 3 4 and 8 7 These are both true statements, but when we add them (left-to-left and right-to-right), we get 3 + 8 4 + 7 But they are equal! We can multiply both sides of a not equal statement by the same nonzero quantity and get another true statement If the quantity is 0, then we end up with 0 0, which is false We can divide both sides of a not equal statement by the same nonzero quantity and get another true statement If the quantity by which we divide through is 0, we get undefined results on both sides of the inequality symbol

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184 Equations and Inequalities

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Here s a summary of how we can morph not equal statements Can we reverse the order Yes Can we add the same quantity to both sides Yes Can we subtract the same quantity from both sides Yes Can we add one statement to another Not in general Can we multiply both sides by the same quantity Only if that quantity is not 0 Can we divide both sides by the same nonzero quantity Yes

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Manipulating > statements If some quantity a is strictly larger than another quantity b, we cannot reverse the order and still have a valid statement It is never true that if a > b, then b > a However, we can reverse the order if we also reverse the sense of the inequality If a > b, then it is always true that b < a We can add or subtract the same quantity from each side of a strictly larger than statement If a > b, then for any number c

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a+c>b+c and a c>b c We can always add two strictly larger than statements (left-to-left and right-to-right) and get another strictly larger than statement For any numbers a, b, c, and d, if we have a > b and c > d, then a+c>b+d We can multiply both sides of a strictly larger than statement by the same positive quantity and get another valid statement If a > b, then for any positive number p ap > bp If the quantity by which we multiply through is 0, then we end up with 0 > 0, which is false If the quantity by which we multiply the statement through happens to be negative, the sense of the inequality is reversed The strictly larger than relation turns into a strictly smaller than relation If a > b, then for any negative number n an < bn We can divide both sides of a strictly larger than statement by the same positive quantity and get another valid statement If a > b, then for any positive number p a /p > b /p

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