By Sheldon Jay Axler

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**Example text**

Should be read to mean “the set of real numbers x such that” and then whatever follows. There is no particular x here. The variable is simply a convenient device to describe a property, and the symbol used for the variable does not matter. Thus {x : x > 2} and {y : y > 2} and {t : t > 2} all denote the same set, which can also be described (without mentioning any variables) as the set of real numbers greater than 2. A special type of set occurs so often in mathematics that it gets its own name, which is given by the following definition.

71. Explain why |ab| = |a||b| for all real numbers a and b. for all real numbers a and b. 76. Show that if a and b are real numbers such that |a + b| < |a| + |b|, 72. Explain why |−a| = |a| for all real numbers a. then ab < 0. 73. Explain why a |a| = b |b| for all real numbers a and b (with b = 0). 74. Give an example of a set of real numbers such that the average of any two numbers in the set is in the set, but the set is not an interval. 77. Show that |a| − |b| ≤ |a − b| for all real numbers a and b.

Simplify the expression 2(3m + x) + 5x. solution First use the distributive property to transform 2(3m + x) into 6m + 2x: 2(3m + x) + 5x = 6m + 2x + 5x. Now use the distributive property again, but in the other direction, to transform 2x + 5x to (2 + 5)x: 6m + 2x + 5x = 6m + (2 + 5)x = 6m + 7x. Putting all this together, we have used the distributive property (twice) to transform 2(3m + x) + 5x into the simpler expression 6m + 7x. 2 Algebra of the Real Numbers 11 One of the most common algebraic manipulations involves expanding a product of sums, as in the following example.