Real Numbers
Real numbers include all numbers: negative, positive, zero, fractions, decimals, most square roots, and so on. Usually, the numbers used on the CBEST will be real numbers, unless otherwise stated.
Variables
Variables are symbols, such as x and y, that are used to replace numbers. The symbol is usually a letter of the alphabet, although occasionally, other symbols are used. When a math problem asks you to "solve for y," that means to figure out what number the letter is replacing. At other times, the problem requires you to work with the letters as if they were numbers. Examples of both will be covered in the lesson on algebra on page 120.
Reciprocal
The reciprocal of a fraction is the fraction turned upside down. For example, the reciprocal of is , and vice versa. The reciprocal of an integer is 1 over the integer. For example, the reciprocal of 2 (or ) is , and vice versa. To get the reciprocal of a mixed number such as , first change the number to an improper fraction and then turn it over .
Numerator and Denominator
The numerator of a fraction is the number on top, and the denominator is the number on the bottom. The numerator of is 6 and the denominator is 7.
= and ≠
The symbol = is called an equal sign. It indicates that the values on both sides of the sign are equal to each other. For example, 7 = 2 + 5. A line drawn through an equal sign (≠) indicates that the values on either side are not equal: 8 ≠ 4 + 5.
< and >; ≤ and ≥
The symbol < means less than, and the symbol > means greater than. The number on the closed side of the symbol is less, and the number on the open side is greater. Thus, 3 < 5 and 10 > 2.Remember: The alligator eats the bigger number.
The symbol ≤ means less than or equal to, and the symbol ≥ means greater than or equal to. These two symbols operate the same way as the < and >, but the added line means that it's possible that the two sides are equal. Thus, in the equation x ≥ 3, x can represent 3 or any number greater than 3.
Answer to Sample Definition Question
Using the definitions you just learned, can you solve sample question 1 from page 96? The variable x can be any whole number including zero. The variable y can be any positive integer, which doesn't include zero. The question reads "…for what value of x MUST x < y be true?" Must means that x has to be less than y under all circumstances, so you are being asked to replace x with a number that will be less than any positive integer that replaces y. The only whole number that would make x < y true, no matter what positive integer is put in place of y, is zero. Therefore, choice b is the correct answer.
Try another sample question. Again, the definitions will be useful in solving this problem.
Sample Digits Question
 In a certain twodigit number, the tens digit is four more than the ones digit. The sum of the two digits is ten. What is the number?
 26
 82
 40
 37
 73
Answer
There are two requirements for the unknown number: The tens digit has to be four more than the ones digit, and the two digits have to add up to 10. The best way to solve the problem is to eliminate answers that don't meet these two requirements. Consider the second criterion first. A glance at the answers shows that the digits in the answers a and c do not add up to 10. They can be eliminated. Next, consider the first requirement. Choice b contains a tens digit that is six, not four, more than the ones digit. Choice d has the ones digit four more than the tens, reversing the requirement. Therefore, choice e is the only number that correctly meets the requirements.
Practice with Definitions
Match the word on the left with the description or example on the right. You may want to write these definitions on flash cards.
Answers
 e.
 c.
 f.
 a.
 b.
 g.
 i.
 h.
 k.
 j.
 d.
 1

2
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