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ALGEBRA CHAPTER 3
Solving and Graphing Linear Inequalities
ONE-STEP LINEAR INEQUALITIES—3.1
VOCABULARY
An equation is formed when an equal sign (=) is
placed between two expressions creating a left and
a right side of the equation
 An equation that contains one or more variables is
called an open sentence
 When a variable in a single-variable equation is
replaced by a number the resulting statement can
be true or false
 If the statement is true, the number is a solution
of an equation
 Substituting a number for a variable in an equation
to see whether the resulting statement is true or
false is called checking a possible solution

INEQUALITIES
Another type of open sentence is called an
inequality.
 An inequality is formed when and inequality sign
is placed between two expressions
 A solution to an inequality are numbers that
produce a true statement when substituted for
the variable in the inequality

INEQUALITY SYMBOLS

Listed below are the 4 inequality symbols and their
meaning
<
Less than
≤
Less than or equal to
>
Greater than
≥
Greater than or equal to
Note: We will be working with inequalities
throughout this course…and you are expected to
know the difference between equalities and
inequalities
GRAPHS OF LINEAR INEQUALITIES

Graph (1 variable)

The set of points on a number line that represents all
solutions of the inequality
GRAPHS OF LINEAR INEQUALITIES
GRAPHS OF LINEAR INEQUALITIES
WRITING LINEAR INEQUALITIES

Bob hopes that his next math test grade will be
higher than his current average. His first three
test scores were 77, 83, and 86.

Why would an inequality be best in this case?

How can we come up with this inequality?

Graph! 
SOLVING ONE-STEP LINEAR INEQUALITIES

Equivalent Inequalities


Two or more inequalities with exactly the same
solution
Manipulating Inequalities

All of the same rules apply to inequalities as
equations*

When multiplying or dividing by a negative number,
we have to switch the inequality!

Less than becomes greater than, etc.
SOLVING WITH ADDITION/SUBTRACTION
SOLVING WITH ADDITION/SUBTRACTION
SOLVING WITH MULTIPLICATION/DIVISION
SOLVING WITH MULTIPLICATION/DIVISION
WHY DO WE HAVE TO CHANGE THE SIGN?

Is there another way we can solve this?
ALGEBRA CHAPTER 3

Solving and Graphing Linear
Inequalities
SOLVING MULTI-STEP LINEAR
INEQUALITIES—3.2
MULTI STEP INEQUALITIES

Treat inequalities just like you would normal,
everyday equations*
*change the sign when multiplying or dividing by a
negative!!
EXAMPLES:
EXAMPLES:
EXAMPLES:
EXAMPLES:
EXAMPLE

You plan to publish an online newsletter that
reports the results of snow cross competitions.
You do not want your monthly costs to exceed
$2370. Your fixed monthly costs are $1200. You
must also pay $130 per month to each article
writer. How many writers can you afford to hire
in a month?
EXAMPLES: TRY THESE ON YOUR OWN!
1) WHICH GRAPH REPRESENTS THE CORRECT
k
ANSWER TO
4
1.
2.
3.
4.
-5
-5
-5
-5
>1
o
-4
o
-4
●
-4
●
-4
Answer Now
-3
-3
-3
-3
x
2) WHEN SOLVING
>
-10
3
WILL THE INEQUALITY SWITCH?
1.
2.
3.
Yes!
No!
I still don’t
know!
Answer Now
3) WHEN SOLVING
a
6
4
WILL THE INEQUALITY SWITCH?
1.
2.
3.
Yes!
No!
I still don’t
know!
Answer Now
4) SOLVE -8P ≥ -96
1.
2.
3.
4.
p ≥ 12
p ≥ -12
p ≤ 12
p ≤ -12
Answer Now
5) SOLVE 7V < -105
1.
2.
3.
4.
o
-16 -15 -14
o
-16 -15 -14
●
-16 -15
●
-14
-15 -15 -14
Answer Now
CLASS WORK:
P.343
#15-37 ODD
IF YOU DO NOT FINISH IN
CLASS, THEN IT BECOMES
HOMEWORK!
ALGEBRA CHAPTER 3

Solving and Graphing Linear
Inequalities
COMPOUND INEQUALITIES—3.6
COMPOUND INEQUALITY

What does compound mean?


Compound fracture?
So…what’s a compound inequality?

An inequality consisting of two inequalities connected
by an and or an or
GRAPHING COMPOUND INEQUALITIES

Graph the following:
GRAPHING COMPOUND INEQUALITIES

Graph the following:
GRAPHING COMPOUND INEQUALITIES

Graph the following:

All real numbers that are greater than or equal to -2
and less than 3
SOLVING COMPOUND INEQUALITIES

Again….treat these like equations!

Whenever we do something to one side…
…We do it to every side!
SOLVING COMPOUND INEQUALITIES
SOLVING COMPOUND INEQUALITIES
SOLVING COMPOUND INEQUALITIES
SOLVING COMPOUND INEQUALITIES
HOMEWORK:
P.349
#12-36 EVEN
SOLVING ABSOLUTE-VALUE EQUATIONS
AND INEQUALITIES—3.6 (DAY 1)
ABS. VALUE

What is Absolute Value?


Distance from zero
What does that mean?
ABS. VALUE


So….an absolute value equation has how many
solutions?
Is this always true?
ABS. VALUE

How do we apply this to equations?

Ex:
EXAMPLES
EXAMPLES
EXAMPLES
EXAMPLES
EXAMPLES
P.356#19-36
SOLVING ABSOLUTE-VALUE EQUATIONS
AND INEQUALITIES—3.6 (DAY 2)
ABSOLUTE VALUE AND INEQUALITIES
ABSOLUTE VALUE AND INEQUALITIES
EXAMPLES
EXAMPLES
EXAMPLES