Real Numbers

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


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








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