**Learning Objectives**

After completing this tutorial, you should be able to:

- Use the addition, subtraction, multiplication, and division properties of inequalities to solve linear inequalities.
- Solve linear inequalities involving absolute values.
- Write the answer to an inequality using interval notation.
- Draw a graph to give a visual answer to an inequality problem.

** Introduction**

In this tutorial we will be looking at solving linear
inequalities.
When solving linear inequalities, we use a lot of the same concepts
that
we use when solving linear equations (shown in **Tutorial
14: Linear Equations in One Variable**). 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. We will also
revisit the definition of absolute value and how it applies to
inequalities.
If you need a review on absolute values go to **Tutorial
21: Absolute Value Equations**. You never know when
you
will need to know about inequalities, so you better get started.

** Tutorial**

*a < b **a* is less than *b*

*a < b *

*a > b **a* is greater than *b*

*a > b*

Interval notation is a way to notate the range of
values that would
make an inequality true. There are two types of intervals, open
and
closed (described below), each with a specific way to notate it so we
can
tell the difference between the two.

Note that in the interval notations (found below), you
will see the
symbol ,
which
means **infinity**.

**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.

**To indicate this, we use a curved end as shown
below. **

**Inequality*** *

*
*

(*a*, )

(-, *a*)

When you graph an open ended end point, you use the
same curved end
( or ) on the graph as you do in the interval
notation.
Also, darken in the part of the graph that is the solution. For
example,

**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**

Closed Intervals

[*a*, )

(-, *a*]

When you graph a closed ended end point, you use the
same boxed end
[ or ] on the graph as you do in the interval
notation.
Also, darken in the part of the graph that is the solution. For
example,

Sometimes one end of your interval is open and the
other end is closed.
You still follow the basic ideas described above. The closed end
will have a [ or ] on it’s end and the open end will have a
( or ) on its end.

Inequality

**Interval Notation for
Combining Open and
Closed Intervals**

(*a*, *b*]

[*a*, *b*)

*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.**

**Interval notation:**

**Graph:**

***Open interval indicating all
values less than
5**

***Visual showing all numbers
less than 5 on
the number line**

Note that the inequality stayed the same throughout the
problem.
Adding or subtracting the same value to both sides does not change the
inequality.

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. **

Note that the inequality stayed the same throughout the
problem.
Adding or subtracting the same value to both sides does not change the
inequality.

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.

when multiplying/dividing by a

*If a < b *AND ** c is positive**, then

*If a < b *AND ** c is positive**, then

**Interval notation: **(-,
-3)

**Graph:**

***Open interval indicating all
values less than
-3**

***Visual showing all numbers
less than -3 on
the number line**

Note that the inequality sign stayed the same
direction. Even
though the right side was a -9, the number we were dividing both sides
by, was a positive 3. **Multiplying or dividing both sides by
the
same positive value does not change the inequality.**

**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.

when multiplying/dividing by a

*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**

I multiplied by a -4 to take care of both the negative
and the division
by 4 in one step.

**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.

This would involve things like removing ( ),
removing fractions, adding
like terms, etc.

**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**

** **

We have an open interval since we are not including where it is equal to -3.

**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.

**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. **

Again, we have an open interval since we are not including where it is equal to 8. This time

**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.

**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. **

This time we have a closed interval since we are including where it is equal to -3/2.

**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.

This is an example of an compound inequality

**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. **

This time we have a mixed interval since we are including where it is equal to -6, but not equal to 8.

**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 there is a constant that is on the same side of the
inequality that
the absolute value expression is but is not inside the absolute value,
use inverse operations to isolate the absolute value.

A quick reminder, the absolute value measures the
DISTANCE a number
is away from the origin (zero) on the number line. No matter if
the
number is to the left (negative) or right (positive) of zero on the
number
line, the DISTANCE it is away from zero is going to be positive. Hence,
the absolute value is always positive, (or zero if you are taking the
absolute
value of 0).

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.

You will solve these linear inequalities just like the
ones shown above.

The absolute value expression is already isolated.

**AND**

**Step 3: Solve
the linear inequalities** **set up in
step
2.**

The distance that the expression* x* - 4
is away from the origin needs to be less than 7.

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**

** **

This time we have an open interval since we are not including either endpoint.

**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.

The absolute value expression is already isolated.

**AND**

**Step 3: Solve
the linear inequalities** **set up in
step
2.**

Be careful, since the absolute value (the left side) is
always positive,
and positive values are always greater than negative values, **the answer
is no solution. **There is no value that we can put in for *x* that would make this inequality true.

***Abs. value exp. isolated**

**AND**

**Step 3: Solve
the linear inequalities** **set up in
step
2.**

The distance that the expression* *(7
- 2*y*)/2
is away from the origin needs to be greater than or equal to 4.

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 - 2*y*)/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 **

This time we have two closed intervals since we are including the endpoints -1/2 and 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.

The absolute value expression is already isolated.

**AND**

**Step 3: Solve
the linear inequalities** **set up in
step
2.**

Again, be careful - since the absolute value (the left
side) is always
positive, and positive values are always greater than negative values, **the
answer is all real numbers. **No matter what value you plug in
for *x*, when you take the absolute value
the
left side will be positive. All positive numbers are greater than
-2.

** Practice Problems**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

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.

Practice Problems 2a - 2d: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.
All rights reserved.