Transcript Inequalities

Inequalities
Objective - TSWBAT Solve simple
inequalities in one variable and
conjunctions and disjunctions.
Inequalities






Properties – Similar to those of Equations from
Chapter 1.
Comparison Property – Exactly only one of the
following statements is true: a < b, a = b, a > b.
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.
Inequalities
Inequality – A sentence formed by placing
an inequality symbol between two
expressions.
 Inequality Symbols – one of the following
symbols: , , , , 
 We do not solve inequalities but transform
them as they do not have set solutions.
 There are five steps to inequalities.

Steps to Transform Inequalities





Step 1 – Simplify by Distributing
Step 2 – Simplify by Combining Like Terms
Step 3 – Add and/or Subtract the same value on
both sides of the inequality to isolate the variable
term.
Step 4 - Multiply and/or Divide the same positive
value on both sides of the inequality to isolate
the variable term.
Step 5 - Multiply and/or Divide the same negative
value on both sides of the inequality to isolate
the variable term and reverse the direction of the
inequality.
Inequalities
 Examples
–
Combined Inequalities
Conjunction – A sentence formed by
joining two inequalities with the word and.
A conjunction is true when both parts of
the sentence are true. If only one
sentence is true the conjunction is false.
 Example: X > b and x < a or a < x < b
 -2 < x < 3 or -2<x and 3>x
 To solve a conjunction you find the values
of the variable for which both parts of the
sentence are true.

Combined Inequalities
 Disjunction
– A sentence formed by
joining two inequalities with the word
or. A disjunction is true when at
least one of the sentences is true.
xb
 Example: x < b or x = b or
 x < 2 or x = 2 or x  2
 To solve a disjunction you find the
values of the variable for which at
least one of the sentences are true.
Combined Inequalities
 Examples
– Conjunctions –
Combined Inequalities
 Examples
– Disjunctions –
Word Problems










Follow the same steps as with equations. They
are:
Step 1 – Read the Problem
Step 2 – Draw a picture/diagram/graph
Step 3 – Define your variable
Step 4 – Label picture/diagram/graph
Step 5 – Re-Read the problem
Step 6 – Set up Equation
Step 7 – Solve
Step 8 – Check your answer
Note about Step 7 – Please make sure you follow
the transformation steps and graph the answer.
Word Problems
 Phrases
to know and their
Translations
Phrase
X
X
X
X
is
is
is
is
at least a.
no less than a.
at most b.
no greater than b.
X is between a and b.
X is between a and b inclusive.
Translation
xa
xb
a<x<b
a xb
Word Problems
 Example
- A bus is to be chartered
for the freshman class trip. The
basic fare is $9.50 per passenger. If
more than 20 people go, everyone’s
fare is reduced by $.30 for each
passenger over this number (20). At
least how many people must go to
make the fare less than $7.50 per
passenger?
 Draw a picture/diagram
Word Problems
 Define
your Variable – x= number of
passengers
 Label your picture
 Re-read problem
 Set-up Equation
 Solve – remember follow rules for
inequalities and graph.
 Check your answer.
Absolute Value and Inequalities




Sentence
Equivalent Sentence
Graph
|a| = 1
a = -1 or a = 1
The distance between a and 0 is 1.
|b| > 1
b < -1 or b > 1
The distance between b and 0 is greater than 1.
|c| < 1
-1 < c < 1
The distance between c and 0 is less than 1.
Absolute Value
 Examples
-
Absolute Value
 Example
– We can also just solve by
graphing. -