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An Introduction to
Inequalities
Presented by J. Grossman
Definition
An inequality is a mathematical
statement that one quantity is
greater than or less than another.
An inequality is like an equation,
but instead of an equal sign (=) it
has one of these signs:
< : less than
≤ : less than or equal to
> : greater than
≥ : greater than or equal to
A solution of an inequality is any real
number that makes the inequality true. For
example, the solutions of the inequality x < 3
are all real numbers that are less than 3.
Can you name them all?
Why? Why not?
x<5
means that whatever
numerical value x has, it
must be less than 5.
Name ten numbers that are
less than 5!
Numbers less than 5 are to the left
of 5 on the number line.
-25 -20 -15 -10 -5
0
5
10 15 20 25
• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.
• There are also numbers in between the integers, like
2.5, 1/2, -7.9, etc.
• The number 5 would not be a correct answer,
though, because 5 is not less than 5.
x ≥ -2
means that whatever value x
has, it must be greater than or
equal to -2.
Name ten numbers that are
greater than or equal to -2!
Numbers greater than -2 are to the
right of -2 on the number line.
-25 -20 -15 -10 -5
0
5
10 15 20 25
-2
• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.
• There are also numbers in between the integers,
like -1/2, 0.2, 3.1, 5.5, etc.
• The number -2 would also be a correct answer,
because of the phrase, “or equal to”.
Where is -1.5 on the number line?
Is it greater or less than -2?
-25 -20 -15 -10 -5
0
5
10 15 20 25
-2
• -1.5 is between -1 and -2.
• -1 is to the right of -2.
• So -1.5 is also to the right of -2.
Three Ways to State an Inequality
• Use inequality notation.
x < –3
x is less than minus three
• Use interval (or set) notation.
{x | x < –3} all x such that x is less than minus three
{x | x is a real number, x < –3}
the interval from minus infinity to minus three
• Graphically display the solution on a
number line.
Graphing Inequalities
x>c
When x is greater than a constant, you darken in the
part of the number line that is to the right of the
constant. Also, because there is no equal line, we
are not including where x is equal to the constant.
That means we are not including the endpoint.
One way to notate that is to use an open circle at
that point on the number line.
Graphing Inequalities
x<c
When x is less than a constant, you darken in the
part of the number line that is to the left of the
constant. Also, because there is no equal line, we
are not including where x is equal to the constant.
That means we are not including the endpoint.
One way to notate that is to use an open circle at
that point on the number line.
Graphing Inequalities
x<c
When x is less than or equal to a constant, you
darken in the part of the number line that is to the
left of the constant. Also, because there is an
equal line, we are including where x is equal to the
constant. That means we are including the
endpoint. One way to notate that is to use a closed
circle at that point on the number line.
Graphing Inequalities
x>c
When x is greater than or equal to a constant, you
darken in the part of the number line that is to the
right of the constant. Also, because there is an
equal line, we are including where x is equal to the
constant. That means we are including the
endpoint. One way to notate that is to use a closed
circle at that point on the number line.
Practice Graphing Inequalities
x  16
x   17
x  54
z   7
5
y 
7
h0
Practice Graphing Inequalities
x is less than 5
y is greater than -3
A number n is positive
b is less than or equal to eight
The speed limit is posted 55 mph. Write
and graph an inequality for this situation.
More practice… state the inequality
that has been graphed.
From the Internet
SpeedMath -- Inequalities
http://education.jlab.org/sminequalit
y/index.html
Online Graphing Calculators
http://webgraphing.com
Any Questions???
Homework…
Pre-Algebra Study Guide Practice 2-8,
all problems.
Algebra Study Guide Practice 3-1,
all problems.
Next step...
Solve Inequalities
Using Addition or Subtraction
Using Multiplication or Division
Reference: CPM Algebra Connections
Solving an Inequality Using
Addition or Subtraction
Solving inequalities that involve addition or subtraction is just like
solving equations that involve addition or subtraction. When solving
linear inequalities, we use a lot of the same concepts that we use when
solving linear equations. Basically, we still want to get the variable on
one side and everything else on the other side by using inverse
operations. The difference is, when a variable is set equal to one
number, that number is the only solution. But, when a variable is less
than or greater than a number, there are an infinite number of values
that would be a part of the answer.
Solving Inequality Using
Addition or Subtraction
Example 1: Solve the following inequality and graph the solution set.
Graph:
Solving an Inequality Using
Addition or Subtraction
Example 2: Solve the following inequality and graph the solution set.
Graph:
Solving an Inequality Using
Addition or Subtraction
Example 3: Solve the following inequality and graph the solution
set.
Graph:
Solving an Inequality Using
Addition or Subtraction
Example 4: Solve the following inequality and graph the solution set.
Solving an Inequality Using
Addition or Subtraction
Example 5: Solve the following inequality and graph the solution set.
4x > 28
A {x | x < 4}
.
B. {x | x < 7}
C. {x | x > 28}
D. {x | x > 7}
Solving an Inequality Using
Addition or Subtraction
Addition/Subtraction Property for Inequalities
If a < b, then a + c < b + c
If a < b, then a - c < b – c
In other words, adding or subtracting the same expression to both
sides of an inequality does not change the inequality.