Learning Objectives
Introduction
Tutorial
Inequality Signs
a < b a is less than b
a < b a is less than or equal to b
a > b a is greater than b
a > b a is
greater than or equal to b
Note that in the interval notations (found below), you
will see the
symbol 
,
which
means infinity.
)
means it goes on and on indefinitely to the right of the number - there
is no endpoint on the right.
Negative infinity (-)
means it goes on and on indefinitely to the left of the number - there
is no endpoint to the left.
Since we don’t know what the largest or smallest numbers are, we need to use infinity or negative infinity to indicate there is no endpoint in one direction or the other.
In general, when using interval notation, you always put the smaller value of the interval first (on the left side), put a comma between the two ends, then put the larger value of the interval (on the right side). You will either use a curved end ( or ) or a boxed end [ or ], depending on the type of interval (described below).
If you have either infinity or negative infinity on either end, you always use a curve for that end. This will indicate that there is no definite endpoint in that direction, it keeps going and going.
Open Interval
To indicate this, we use a curved end as shown below.
Inequality
) , a) )
x < 4
(-, 4)
To indicate this, we use a boxed end as shown below.
As mentioned above, even though a is included and has a boxed end, if it goes to either infinity or negative infinity on the other end, we will notate it with a curved end for that end only!
Inequality
Interval Notation for
[a, )
, a])
x < 4
(-, 4]
Inequality
Interval Notation for
Combining Open and
Closed Intervals
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.
Example
1: Solve, write your answer in interval notation and
graph
the solution set:
.
Interval notation:
Graph:
*Open interval indicating all values less than 5
*Visual showing all numbers less than 5 on the number line
The answer 'x is less than 5' means that if we put any number less than 5 back in the original problem, it would be a solution (the left side would be less than the right side). As mentioned above, this means that we have more than just one number for our solution, there are an infinite number of values that would satisfy this inequality.
Interval notation:
We have an open interval since we are not including where it is equal
to 5. x is less than
5, so
5 is the largest value of the interval, so it goes on the right.
Since there is no lower endpoint (it is ALL values less than 5), we put
the negative infinity symbol on the left side. The curved end on
5 indicates an open interval. Negative infinity always has a
curved
end because there is not an endpoint on that side.
Graph:
We use the same type of notation on the endpoint as we did in the
interval
notation, a curved end. Since we needed to indicate all
values
less than 5, the part of the number line that was to the left of 5 was
darkened.
Example 2: Solve, write your answer in interval notation and graph
the
solution set: 
.
Interval notation: [-3, )
Graph:
*Closed interval indicating all values greater than or = -3
*Visual showing all numbers greater than or = to -3 on the number line.
The answer 'x is greater than or equal to -3' means that if we put any number greater than or equal to -3 back in the original problem, it would be a solution (the left side would be greater than or equal to the right side). As mentioned above, this means that we have more than just one number for our solution, there are an infinite number of values that would satisfy this inequality.
Interval notation:
We have a closed interval since we are including where it is equal
to -3. x is greater than or
equal
to -3, so -3 is our smallest value of the interval, so it goes on the
left.
Since there is no upper endpoint (it is ALL values greater than or
equal
to -3), we put the infinity symbol on the right side. The boxed
end
on -3 indicates a closed interval. Infinity always has a curved
end
because there is not an endpoint on that side.
Graph:
We use the same type of notation on the endpoint as we did in the
interval
notation, a boxed end. Since we needed to indicate
all
values greater than or equal to -3, the part of the number line that
was
to the right of -3 was darkened.
If a < b AND c is positive, then ac < bc
If a < b AND c is positive, then a/c < b/c
Example
3: Solve, write your answer in interval notation and
graph
the solution set: 
.
Interval notation: (-,
-3)
Graph:
*Open interval indicating all values less than -3
*Visual showing all numbers
less than -3 on
the number line
Interval notation:
We have an open interval since there we are not including where it
is equal to -3. x is less
than
-3, so -3 is our largest value of the interval, so it goes on the
right.
Since there is no lower endpoint (it is ALL values less than -3), we
put
the negative infinity symbol on the left side. The curved end on
-3 indicates an open interval. Negative infinity always has a
curved
end because there is not an endpoint on that side.
Graph:
We use the same type of notation on the endpoint as we did in the
interval
notation, a curved end. Since we needed to indicate all
values
less than -3, the part of the number line that was to the left of -3
was
darkened.
If a < b AND c is negative, then ac > bc
If a < b AND c is negative, then a/c > b/c
The reason for this is, when you multiply or divide an
expression by
a negative number, it changes the sign of that expression. On the
number line, the positive values go in a reverse or opposite direction
than the negative numbers go, so when we take the opposite of an
expression,
we need to reverse our inequality to indicate this.
Example
4: Solve, write your answer in interval notation and
graph
the solution set: 
.
Interval notation: 
Graph:
*Open interval indicating all values less than -20
*Visual showing all numbers
less than -20 on
the number line
In line 2, note that when I did show the step of multiplying both sides by a -4, I reversed my inequality sign.
Interval notation:
We have an open interval since we are not including where it is equal
to -20. x is less than
-20,
so -20 is our largest value of the interval, so it goes on the
right.
Since there is no lower endpoint (it is ALL values less than -20), we
put
the negative infinity symbol on the left side. The curved end on
-20 indicates an open interval. Negative infinity always has a
curved
end because there is not an endpoint on that side.
Graph:
We use the same type of notation on the endpoint as we did in the
interval
notation, a curved end. Since we needed to indicate all
values
less than -20, the part of the number line that was to the left of -20
was darkened.
Example
5: Solve, write your answer in interval notation and
graph
the solution set: 
.
Interval notation: 
Graph:
*Closed interval indicating all values greater than or = -5/2
*Visual showing all numbers
greater than or
= -5/2 on the number line
Interval notation:
We have a closed interval since we are including where it is equal
to -5/2. x is greater than
or equal
to -5/2, so -5/2 is our smallest value of the interval so it goes on
the
left. Since there is no upper endpoint (it is ALL values greater
than or equal to -5/2), we put the infinity symbol on the right
side.
The boxed end on -5/2 indicates a closed interval. Infinity
always
has a curved end because there is not an endpoint on that side.
Graph:
We use the same type of notation on the endpoint as we did in the
interval
notation, a boxed end. Since we needed to indicate
all
values greater than or equal to -5/2, the part of the number line that
was to the right of -5/2 was darkened.
Step 2: Use Add./Sub. Properties to move the variable term on one side and all other terms to the other side.
Step 3: Use Mult./Div.
Properties to remove any values that are in front of the variable.
Note that it is the same basic
concept we used
when solving linear equations as shown in Tutorial
14: Linear Equations in One Variable.
Example
6: Solve, write your answer in interval notation and graph
the
solution set: 
.
Interval notation:
Graph:
*Inv. of mult. by -2 is div.
both sides by
-2, so reverse inequality sign
*Open interval indicating all
values greater
than -3
*Visual showing all numbers
greater than -3
on the number line
Graph:
We use the same type of notation on the endpoint as we did in the
interval
notation, a curved end. Since we needed to indicate all
values
greater than -3, the part of the number line that was to the right of
-3
was darkened.
Example
7: Solve, write your answer in interval notation and
graph
the solution set: 
.
Interval notation:
Graph:
*Inv. of sub. 3 is add. by 3
*Open interval indicating all
values less than
-1/2
*Visual showing all numbers less than -1/2 on the number line.
Graph:
Again, we use the same type of notation on the endpoint as we did in
the interval notation, a curved end. Since we needed to indicate
all values less than 8, the part of the number line that was to the
left
of 8 was darkened.
Example
8: Solve, write your answer in interval notation and
graph
the solution set: 
.
Interval notation:
Graph:
*Get x terms on
one side, constants on the other side
*Inv. of add. 3 is sub. by 3
*Inv. of mult. by 10 is div. by
10
*Closed interval indicating all
values greater
than or equal to -3/2
*Visual showing all numbers greater than or equal to -3/2 on the number line.
Graph:
Again, we use the same type of notation on the endpoint as we did in
the interval notation, a boxed end this time. Since we
needed
to indicate all values greater than or equal to -3/2, the part of the
number
line that was to the right of -3/2 was darkened.
Example
9: Solve, write your answer in interval notation and
graph
the solution set: 
.
Interval notation:
Graph:
*All values between -6 and 8,
with a closed
interval at -6 (including -6)
*Visual showing all numbers between -6 and 8, including -6 on the number line.
Graph:
Again, we use the same type of notation on the endpoints as we did
in the interval notation, a boxed end on the left and a curved
end
on the right. Since we needed to indicate all values between -6
and
8, including -6, the part of the number line that is in between -6 and
8 was darkened.
If you need a review on absolute values, feel free to go to Tutorial 21: Absolute Value Equations.
-d < x < d
The graph below illustrates all the values on the number
line whose
distance would be less than d units away
from
0. It shows us why we set up the inequality, shown above, the way
we do.
there is no solution
The absolute value is always positive, and any positive number is greater than any negative number, therefore it would be no solution.
x < -d OR x > d
The graph, shown below, illustrates all the values on
the number line
whose distance would be greater than d units
away from 0. It shows us why we set up the inequality, shown
above,
the way we do.
x is all real numbers
The absolute value is always positive, and any positive number is greater than any negative number, therefore all real numbers would work.
Example
10: Solve, write your answer in interval notation and
graph the solution set: 
.
AND
Step 3: Solve the linear inequalities set up in step 2.
All numbers between -7 and 7 are less than 7 units away from the origin. So, the expression x - 4 needs to be between -7 and 7.
Interval notation:
Graph:
*All values between -3 and 11
*Visual showing all numbers
between -3 and
11
Graph:
Again, we use the same type of notation on the endpoints as we did
in the interval notation, a curved end on both ends. Since
we needed to indicate all values between -3 and 11, the part of
the
number line that is in between -3 and 11 was darkened.
Example
11: Solve, write your answer in interval notation and
graph the solution set: 
.
AND
Step 3: Solve the linear inequalities set up in step 2.
Example
12: Solve, write your answer in interval notation and
graph the solution set: 
.
*Abs. value exp. isolated
AND
Step 3: Solve the linear inequalities set up in step 2.
All numbers that are less than or equal to - 4 OR greater than or equal to 4 are greater than or equal to 4 units away from the origin. So the expression (7 - 2y)/2 needs to be less than or equal to - 4 OR greater than or equal to 4.
OR
Interval notation:
Graph:
*Inv. of div. by 2 is mult. by 2
*Inv. of mult. by -2 is div. by
-2, so reverse inequality sign
*Second inequality, where it is greater than or = to 4
*Inv. of div. by 2 is mult. by 2
*Inv. of mult. by -2 is div. by
-2, so reverse inequality signs
*All values less than or = to
-1/2 or greater
than or = to 15/2
*Visual showing all numbers less than or = to -1/2 or greater than or = to 15/2
In the first interval, y is less than or equal to -1/2, so -1/2 is our largest value of the interval so it goes on the right. Since there is no lower endpoint of that first interval, we put negative infinity on the left side. The boxed end on -1/2 indicates a closed interval. Infinity always has a curved end because there is not an endpoint on that side.
In the second, interval, y is greater than or equal to 15/2, so 15/2 is our smallest value of the interval so it goes on the left. Since there is no upper endpoint of that second interval, we put the infinity symbol on the right side. The boxed end on 15/2 indicates a closed interval. Infinity always has a curved end because there is not an endpoint on that side.
Graph:
Again, we use the same type of notation on the endpoints as we did
in the interval notation, a boxed end on both y = -1/2 and y = 15/2. Since we
needed
to indicate all values less than or equal to -1/2 OR greater than or
equal
to 15/2, the parts of the number line that are to the left of -1/2 and
to the right of 15/2 were darkened.
Example
13: Solve, write your answer in interval notation and
graph the solution set: 
.
AND
Step 3: Solve the linear inequalities set up in step 2.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: Solve, write your answer in interval notation and graph the solution set.
Need Extra Help on these Topics?
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut10_linineq.htm
This website helps you with linear inequalities.
http://www.sosmath.com/algebra/inequalities/ineq01/ineq01.html
This website helps you with linear inequalities.
http://www.math.com/school/subject2/lessons/S2U3L4DP.html
This website helps you with linear inequalities.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 17, 2009 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward.
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