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Linear Equations and Inequalitie
Solving Inequalities
Absolute Value Inequalities
Graphing Equivalent Fractions Lesson Plan
Investigating Liner Equations Using Graphing Calculator
Graphically solving a System of two Linear Equatio
Shifting Reflecting Sketching Graph
Graphs of Rational Functions
Systems of Equations and Inequalities
Graphing Systems of Linear Equat
LINEAR FUNCTIONS: SLOPE, GRAPHS AND MODELS
Solving Inequalities with Absolute Values
Solving Inequalities
Solving Equations & Inequalities
Graph the rational function
Inequalities and Applications
Inequalities
Using MATLAB to Solve Linear Inequalities
Equations and Inequalities
Graph Linear Inequalities in Two Variables
Solving Equations & Inequalities
Teaching Inequalities:A Hypothetical Classroom Case
Graphing Linear Inequalities and Systems of Inequalities
Inequalities and Applications
Solving Inequalities
Quadratic Inequalities
Inequalities
Solving Systems of Linear Equations by Graphing
Systems of Equations and Inequalities
Graphing Linear Inequalities
Inequalities
Solving Inequalities
Solving Inequalities
Solving Equations Algebraically and Graphically
Graphing Linear Equations
Solving Linear Equations and Inequalities Practice Problems
Graphing Linear Inequalities
Equations and Inequalities
Solving Inequalities

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Inequalities and Applications

■ Solving Inequalities

■ Interval Notation

■ The Addition Principle for Inequalities

■ The Multiplication Principle for Inequalities

■ Using the Principles Together

■ Problem Solving

Solving Inequalities
An _________ is any sentence with<, >, ≤, ≥ , or ≠ .

Examples
3x + 2 > 7, c ≤ 7, and 4x − 6 ≠ 3.
Solving Inequalities
A _______ is any value for a variable that makes an inequality
____ . The set of all solutions is called the .

Examples

Determine whether Graph x < 2 and
5 is a solution to express in set builder notation
3x + 2 > 7  

Interval Notation
An _________ expresses a set of numbers. They are written:
•_________________________
•_________________________ the end points from the set
Interval Notation
•_________________________ the end points from the set.
If a and b are real numbers such that a < b:
The open interval (a, b) is the set of all x’s graphed
The closed interval [a, b] is the set of all x’s graphed
The half-open interval (a, b] is the set of all x’s graphed
The interval [a, ∞ ) is the set of all x’s graphed
The interval (-∞, a) is the set of all x’s graphed

Graphs, Sets and Intervals - Practice
Graph:
1. (-2, 4]
Write each in set notation:
1. (-2, 4]

Write each in interval notation:
1. {x | 1 < x < 7}

The Addition Principle for Inequalities

For any real numbers a, b, and c:

a < b a ≥ b
is equivalent to is equivalent to
a + c < b + c; a + c ≥ b + c;

Example
Solve and graph x – 2 > 7.

The Multiplication Principle for Inequalities

For any real numbers a, b, and for any positive number c
a ≤ b a> b
is equivalent to is equivalent to
a·c ≤ b·c; a·c > b·c;
For any real numbers a, b, and for any negative number c
a < b  ______________
 ______________
 ______________
is equivalent to
a·c > b·c;

Solve and graph, also write solution in set and interval notation.
−2(x − 5) − 3x ≤ 4x − 7

Problem Solving

Translation
Tom’s construction work can be paid to him in two ways:
Plan A: $300 plus $9 per hour or
Plan B: Straight $12.50 per hour.
If the job takes n hours,
what is the value of n so that Plan B is better for Tom.

Intersections, Unions, and
Compound Inequalities

■ Intersection of Sets and Conjunctions of
4.2
Sentences

■ Unions of Sets and Disjunctions of Sentences

■ Interval Notation and Domains

Intersection and Conjunction
The ____________ _ of two sets
A and B is the set of all elements
that are common to both A and B.
Intersection is written as ______

A _______________is two or more
sentences are joined by _________.
Ex: a conjunction of inequalities

Example:
Solve and graph and state answer as an interval:
2x +1≥ −3 and −3x > −12

Mathematical Use of “and”

________________________________________
of these ideas to the symbol ___
Any solution of an intersection must make every part true

Note that for a < b,
_______________ b > x and x > a
______________ can be abbreviated
______________ b > x > a
3 < 2x +1 < 7 is the same as 3 < 2x +1 and 2x + 1 < 7

Union and Disjunction
The of two sets A or B
is the combination of all elements
contained in both A and/or B.
Union is written as _________

A is two or more
sentences are joined by _______.
Ex: a disjunction of inequalities

Mathematical Use of “or”

_______________________________________
of these ideas to the symbol ___
A solution to a union makes part of the it true

Example Solve and graph:
2x +1≥ 3 or 3x < −3.