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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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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} 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.