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Chapter 2
Inequalities and Proof
Section 2-1
Solving Inequalities in
One Variable
Properties of Order
Transitive Property - If a < b
and b < c, then a < c
Addition Property - If a < b,
then a + c < b + c
Multiplication Property
1. If a < b and c is positive,
then ac < bc
2. If a < b and c is negative,
then ac > bc
Equivalent Inequalities
Inequalities with the same
solution set
2x + 5 < 13 and 2x < 8 and x < 4
4x > 2(3 + 2x) and 2x > 3 + 2x
Transformations that Produce
Equivalent Inequalities
Simplifying either side of an
inequality.
Transformations that Produce
Equivalent Inequalities
Adding to (or subtracting
from) each side of an
inequality, the same
number or the same
expression.
Transformations that Produce
Equivalent Inequalities
Multiplying (or dividing)
each side of an inequality
by the same negative
number and reversing the
inequality.
Transformations that Produce
Equivalent Inequalities
Multiplying (or dividing)
each side of an inequality by
the same positive number
Examples
Solve each inequality and
graph its solution set
5x + 17 < 2
 5(3-t) < 7 - t

Section 2-2
Solving Combined
Inequalities
ConjunctionA sentence formed by joining two
sentences with the word and. In a
conjunction both sentences are
true.
Example:
Graph the solution set of the
conjunction x > -2 and x < 3
DisjunctionA sentence formed by joining
two sentences with the word or.
It is true when at least one of the
sentences is true.
Example:
Graph the solution set for the
disjunction x < 2 or x = 2
Conjunctions in a Different form
Solve 3 < 2x + 5 ≤ 15.
First rewrite the conjunction with
and.
3 < 2x + 5 and 2x + 5 ≤ 15
Now solve each inequality and
graph the solution set for the
conjunction.
Conjunctions in a Different form
Solve -3 < 2(t -3) < 6.
First rewrite the conjunction with
and.
-3 < 2(t-3) and 2(t-3) < 6
Now solve each inequality and
graph the solution set for the
conjunction.
Disjunctions
Solve 2t + 7  13 or 5t – 4 < 6.
Now solve each inequality and
graph the solution set for the
conjunction.
2t + 7  13 or 5t – 4 < 6
Disjunctions
Solve y  -1 or y  3
Now solve each inequality and
graph the solution set for the
conjunction.
y  -1 or y  3
Section 2-3
Problem Solving Using
Inequalities
Solving Word Problems
Using Inequalities
Phrase
Translation
x is at least a
x is no less than a
x≥a
x is at most b
x is no greater than b
x≤b
x is between a and b
x is between a and b,
inclusive
a<x<b
a≤x≤b
Example:
Find all sets of 4 consecutive
integers whose sum is between 10
and 20.
Solution
Four consecutive integers –
n + (n + 1) + (n + 2) + (n + 3)
 10 < 4n + 6 < 20
1 < n < 14/4
Which integers work?
Section 2-4
Absolute Value in Open
Sentences
Absolute Value
The distance between a number x
and zero on a number line
If | x | = 1, then x = 1 or -1
If | x | < 1, then -1 < x < 1
If | x | > 1, then x < -1 or x > 1
Example - Equality
Solve |3x - 2| = 8
To solve, set up two equations

3x – 2 = -8
3x – 2 = 8
3x = -6
3x = 10
x = -2
x = 10/3
The solution is {-2, 10/3}
Example - Inequality
Solve |3 – 2t| < 5
Set up a compound inequality

-5 < 3 – 2t < 5
-8 < -2t < 2
4>t>-1
The solution set is { t: -1 < t < 4}
Section 2-5
Solving Absolute Value
Sentences Graphically
Facts
The distance between x and
0 on a number line is | x |
The distance between the
graphs of real numbers a
and b is | a – b |, or | b – a |
Examples

Solve |5 - x| = 2
{3, 7}
Examples
Solve |b + 5| > 3
{b: b < -8 or b > -2}
Examples
 Solve
|2n + 5| ≤ 3
{n: n ≤ -4 or n ≥ -1}
Section 2-6
Theorems and Proofs
Definitions
 Theorem - A statement that can be
proved
 Corollary – A theorem that can be
proved easily from another
 Axioms – Statements that we
assume to be true (these are also
called postulates)
Cancellation Property of Addition
For all real numbers a, b, and c:
If a + c = b + c, then a = b
If c + a = c + b, then a = b
Cancellation Property of
Multiplication
For all real numbers a and b, and
nonzero real numbers c:
If ac = bc, then a = b
If ca = cb, then a = b
Zero – Product Property
 For all real numbers a and b:
ab = 0 if and only if a = 0 or b = 0
Section 2-7
Theorems about Order and
Absolute Value