By Sheldon Jay Axler

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15 < 21 5 c Transitivity: a < b and b < c implies that a < c. Often multiple inequalities are written together as a single string of inequalities. Thus a < b < c means the same thing as a < b and b < c. Our next result shows that we can add inequalities. Addition of inequalities If a < b and c < d, then a + c < b + d. To see why this is true, note that if a < b and c < d, then b − a and d − c are positive numbers. Because the sum of two positive numbers is positive, this implies that (b − a) + (d − c) is positive.

Neither ∞ nor −∞ is a real number; these symbols have no meaning in this context other than as notational shorthand. For example, the interval (2, ∞) is defined to be the set of real numbers greater than 2 (note that ∞ is not mentioned in this definition). The notation (2, ∞) is often used because writing (2, ∞) is easier than writing {x : x > 2}. As before, a parenthesis indicates that the corresponding endpoint is not included in the set, and a straight bracket indicates that the corresponding endpoint is included in the set.

We use the identity above (thinking of x as a), which shows that dividing by Here the size of the fraction bars are used to indicate that c . b Thus we have = y c · x b = yc . xb Finally, we conclude this subsection by recording some identities involving fractions and additive inverses: When faced with complicated expressions involving fractions that are themselves fractions, remember that division by a fraction is the same as multiplication by the fraction flipped over. 24 chapter 1 The Real Numbers Fractions and additive inverses −a a a = =− b −b b −a a = −b b a c ad − bc − = b d bd Symbolic Calculators The symbol ≈ means “approximately equal”.