Real Numbers

Real Numbers
Types of Real Numbers ? 1. Natural numbers
2.Integer numbers
3.Rational numbers

Examples

4.Irrational Numbers: These are numbers that cannot be expressed as a ratio
of integers.
Examples






Intervals
Certain sets of real numbers, called intervals, occur frequently in calculus and correspond geometrically to line segments. If , then the open interval from to consists of all numbers between and and is denoted by the symbol . The closed interval from to includes the endpoints and is denoted . Using set-builder notation, we can write

The following table lists the possible types of intervals



Absolute Value: The absolute value of a number , denoted by


Inequalities:
An inequality is an expression involving one of the symbols
When we are asked to solve an inequality, the inequality will contain an unknown variable, say Solving means obtaining all values of x for which the inequality is true.
Properties:
1. Adding or subtracting the same quantity from both sides of an inequality leaves the inequality sign unchanged.

2. Multiplying or dividing both sides by a positive number leaves the inequality sign unchanged.

3. Multiplying or dividing both sides by a negative number reverses the inequality.

Example 1: Solve

Example 2: Solve


Example 3: Solve




Example 4: Solve



The solution is

Solve quadratic inequality:
A quadratic inequality can be written in one of the standard forms


To solve a quadratic inequality in one variable, we will use the following steps to find the values of the variable that make the inequality true.
1- Write the inequality in standard form and solve its related quadratic equation.
2- Locate the solutions ( called critical numbers ) of the related quadratic equation on a number- line.
3- Test each interval on the number line created in step 2 by choosing a test value from the interval and determining whether it satisfies the inequality. The solution set includes the intervals whose test value make the inequality true.
4- Determine whether the endpoints of the intervals are included in the solution set.
Example 5: Solve
Solution:
We will solve the related quadratic equation

These two critical numbers will separate the number-line into three intervals


If we choose
If we choose
If we choose
Then the solution is the interval
Example 6: Solve
Solution: We will solve the related quadratic equation

These two critical numbers will separate the number-line into three intervals

If we choose
If we choose
If we choose
Then the solution is
Example 7: Solve
Solution:
The zero of numerator is
The zero of denominator is
These two critical numbers will separate the number-line into three intervals




Then the solution is

Solve the inequalities