Real Numbers (R) This is the union of rational and irrational numbers.

Numbers that are either rational or irrational are called real numbers.

Rational Numbers (Q)

Any number that can be written in the form of a / b, where a and b are elements of the Integers, but b is not equal to zero.

Rational numbers are generally expressed using fractional or decimal notation. Those numbers for which a decimal notation either ends (terminates) or repeats. Converting decimals to fractions. All terms within the decimal area of the rational number must be identical.

Integers ( I )

I = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

The set of all whole numbers and their opposites. (All positive and negative numbers)
The dots mean that the pattern continues in the direction indicated.

The integers correspond to the points on a number line as follows:

7, and - 5 are integers as they are positive and negative whole numbers. 0.75 or 12 / 5 are not integers as they are decimals or fractions of a whole number, the decimals or fractions are found between integers.

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Whole Numbers ( W or N_o)

W = {0, 1, 2, 3, 4, 5, 6, ...}   Notice the extra zero. These are also called natural numbers with zero included.

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Natural Numbers ( N )

N = {1, 2, 3, 4, 5, 6, ...}

These numbers have been studied for thousands of years and were the first numbers used by mathematicians.

l.c.m. = lowest common multiple, smallest number both numbers will go into.

Example - the l.c.m. of 4 and 5 is 20.

g.c.f. = greatest common factor. largest number which is a factor of both numbers.

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Irrational Numbers

These numbers are found on the top of the number line below. Irrational numbers can only be approximated by rational numbers.

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Absolute Values

We write | a |, read "the absolute value of a," to represent the number of units that a is from zero on a real number line.

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Example 1.

| - 3 | = 3, - 3 is 3 units from 0.

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Example 2.

| 7.2 | = 7.2 is 7.2 units from 0.

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Example 3.

| 0 | = 0 is 0 units from itself.
zero is neither positive or negative, it just an absolute value.

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Example 4.

| 5 - 6 | = | - 1 | = 1 is 1 units from 0.

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Example 5.

| 2 | - | - 5 | = 2 - 5 = - 3

Note that whereas the absolute value of a nonnegative number is the number itself, the absolute value of a negative number is positive.

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Solving Inequalities - An inequality is any sentence having one of the verbs <, >, , or .

Some replacements for an inequality make it true and some make it false. A replacement that makes it true is called a solution. The set of all solutions is called the solution set. When we have found the set of all solutions of an inequality, we say that we have solved the inequality.

We will also find it necessary to have a notation for how any two real numbers compare with each other. For any two numbers on the number line, the one to the left is said to be less than, or smaller than, the one to the right. The symbol < mean "is less than" and the symbol > means "greater than." The symbol means "less than or equal to" and the symbol means "greater than or equal to." These symbols are used to form inequalities.

Generally a < b means that a is less than b and is true when a is to the left of b on the number line. The sentence a > b means that a is greater than b and is true when a is to the right of the b on the number line. Thus, - 3 < 5 is true and 7 > 0 is true. If a < b is true, the b > a is also true. Thus, - 3 < 5 and 5 > - 3 are both true.

In the figure below, note that although | - 3 | > | - 1 |, we have - 3 < - 1 since - 3 is to the left of - 1.

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Example 6.

- 7 < - 2

- 7 is less than - 2, a true statement since - 7 is to the left of - 2.

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Example 7.

4 > - 1

4 is greater than - 1, a true statement.

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Example 8.

3 is greater than (or equal to) 2. This statement is true.

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Example 9.

5 is less than or equal to 6. Since 5 < 6 is true, is true.

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Example 10.

6 is less than or equal to 6. Since 6 = 6 is true, is true.

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Graphing on a Real Number Line

A graph of an inequality is a drawing that represents its solution. An inequality in one variable can be graphed on a number line. Inequalities in two variables can be graphed on a coordinate plane (Cartesian Plane).

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Example 11.

Graph x < 4 on a number line.

The solutions are all real numbers less than 4, so we shade all numbers less than 4. Note that 4 is not a solution. We indicate this by using an open circle at 4.

The solution set can be named as follows, { x | x < 4}.

This is read

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Example 12.

Graph y - 2 on a number line on a number line and write the solution set.

The solution set can be named as follows, { y | y - 2}.

To graph the solution, we shade all the numbers to the right of - 2 and use a solid circle at - 2 to indicated that it is also a solution.

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Example 13.

Solve x + 5 > 3. Then graph.

Any number greater than - 2 is a solution. Thus the solution set is, { x | x > - 2},

and the graph of the inequality is as follows:

We cannot check all the solutions of an inequality by substitution, as we can check solutions of equations. There are too many of them. A partial check could be done by substituting a number greater than - 2, say - 1, into the original inequality:

Since 4 > 3 is true, - 1 is a solution. Any number greater than - 2 is a solution.

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Example 14.

Solve 4x - 1 5x - 2. Then graph.

We know that 1 x has the same meaning as x 1. Thus any number less than or equal to 1 is a solution. We can express the solution set as { x | 1 x}. The graph of the inequality is as follows:

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Example 15.

Solve - 4x < (4 / 5). Then graph.

Any number greater than (- 1 / 5) is a solution. The solution set is { x | x > - 1 / 5}. The graph of the inequality is as follows:

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Example 16.

Solve - 5x - 80. Then graph.

The solution set is { x | x 16}. The graph of the inequality is as follows:

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Example 17.

Solve | - 1 | < x < | - 4 |

The solution set is { x | 1 < x < 4}. The graph of the inequality is as follows:

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Example 18.

Solve - 3 (x + 8) - 5x < 4 (x - 9) + 27.

The solution set is { x | (- 5 / 4) <x}.