# Inequalities

Definitions

Rational Numbers: A number that can be expressed as the quotient of two integers
Examples of Rational Numbers: 1, 5, -4, -8,2/3,-4/3, 3.45, 2.346
Irrational Numbers: A number that can not be expressed as the quotient of two
integers,
Examples of Irrational Numbers:

Example 1 Identify each number as rational or irrational
a) - 343 ( This is a rational number)
b) (This number is irrational)
c) 24/7 (This number is rational)
d) 2e (This number is irrational)

Inequalities
Symbols

> Greater than
≥ Greater than or equal to
< Less than
≤ Less than or equal to

The number line

Solving Inequalities

Inequality Properties

Transitive Property a > b and b > c a > c

Addition Property for Inequalities a > b a + c > b + c

Multiplication properties for Inequalities

(If c is positive, then a > b ac > bc, c > 0 )
(If c is negative, then a > b ac < bc, c < 0 )

Subtraction property for Inequalities a > b a − c > b − c

Example 2

Solve the following inequality 2x + 4 >10

simplify
divide by 2
simplify

Example 3

Solve the following inequality 12x + 36 ≤ 6x + 48

subtract x from both sides
simplify
substract 36 from both sides

divide by 6

Example 4

Solve the following inequality

Compound Inequalities

Example 5

Solve

Example 6

Solve

Example 7

Solve the following inequality 0 ≤ x + 3 ≤ 5

Example 8

Solve the following inequality − 4 ≤ 3x + 3 ≤ 5