Beginning Algebra
Tutorial 18: Solving Linear Inequalities
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.
 Draw a graph to give a visual answer to an inequality problem.

Introduction
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. 
Tutorial
Read left to right:
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 
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 hole at that
point.

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 hole at that point.

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 an closed hole at
that point.

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 hole at that point.


Example
1: Graph x > 5. 
Since we needed to indicate all values greater than 5, the part
of the number line that is to the right of 5 was darkened.
Since there is no equal line under the > symbol, this means we do not
include the endpoint 5 itself. We can notate that by using an open
hole (or you can use a curved end). 
Example
2: Graph x < 2. 
Since we needed to indicate all values less than or equal to 2,
the part of the number line that is to the left of 2 was darkened.
Since there is an equal line under the < symbol, this means we do
include the endpoint 2. We can notate that by using a closed hole
(or you can use a boxed end). 
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.
Example
3: Solve the inequality and graph the solution set. 
Graph:

*Inv. of sub. 7 is add. 7
*Visual showing all numbers less than 4 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 4' means that
if we put any number less than 4 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.
Graph:
Since we needed to indicate all values less than 4, the part of the
number line that was to the left of 4 was darkened.
Since we are not including where it is equal to, an open hole was used. 
Example
4: Solve the inequality and graph the solution set. 
Graph:

*Inv. of add 10 is sub. 10
*Visual showing all numbers greater than or
= to 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 greater than or equal to
5' means that if we put any number greater than or equal to 5 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.
Graph:
Since we needed to indicate all values greater than or equal to 5,
the part of the number line that was to the right of 5 was darkened.
Since we are including where it is equal to, a closed hole was used. 
Multiplication/Division Properties for Inequalities
when multiplying/dividing by a positive value
If a < b AND c is positive, then
ac < bc
If a < b AND c is positive, then
a/c < b/c

In other words, multiplying or dividing the same POSITIVE number
to both sides of an inequality does not change the inequality.
Example
5: Solve the inequality and graph the solution set. 
Graph:

*Inv. of mult. by 5 is div. by 5
*Visual showing all numbers less than 2 on
the number line

Note that the inequality sign stayed the same direction. Even
though the right side was a 10, the number we were dividing both sides
by, was a positive 5. Multiplying or dividing both sides by the
same positive value does not change the inequality.
Graph:
Since we needed to indicate all values less than 2, the part of the
number line that was to the left of 2 was darkened.
Since we are not including where it is equal to, an open hole was used. 
Example
6: Solve the inequality and graph the solution set. 
Graph:

*Inv. of div. by 3 is mult. by 3
*Visual showing all numbers greater than 3
on the number line

Multiplying or dividing both sides by the same positive value does
not change the inequality.
Graph:
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.
Since we are not including where it is equal to, an open hole was used. 
Multiplication/Division Properties for Inequalities
when multiplying/dividing by a negative value
If a < b AND c is negative, then
ac > bc
If a < b AND c is negative, then
a/c > b/c

In other words, multiplying or dividing the same NEGATIVE number
to both sides of an inequality reverses the sign of the inequality.
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
7: Solve the inequality and graph the solution set. 
Graph:

*Inv. of div. by 2 is mult. by 2, so reverse inequality sign
*Visual showing all numbers less than 14 on
the number line

I multiplied by a 2 to take care of both the negative and the division
by 2 in one step.
In line 2, note that when I did show the step
of multiplying both sides by a 2, I reversed my inequality sign.
Graph:
Since we needed to indicate all values less than 14, the part of the
number line that was to the left of 14 was darkened.
Since we are not including where it is equal to, an open hole was used. 
Example
8: Solve the inequality and graph the solution set. 
Graph:

*Inv. of mult. by 3 is div. by 3, so reverse inequality sign
*Visual showing all numbers greater than or
= 3 on the number line

In line 2, note that when I did show the step
of dividing both sides by a 3, that I reversed my inequality sign.
Graph:
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.
Since we are including where it is equal to, a closed hole was used. 
Strategy for Solving a Linear Inequality

Step 1: Simplify each side, if needed.
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: Solving Linear Equations.
Example
9: Solve the inequality and graph the solution set. 
Graph:

*Inv. of sub. 3 is add. 3
*Inv. of mult. by 3 is div. both sides by
3, so reverse inequality sign
*Visual showing all numbers greater than 3
on the number line

Graph:
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.
Since we are not including where it is equal to, an open hole was used. 
Example
10: Solve the inequality and graph the solution set. 
Graph:

*Distributive property
*Get x terms on one side, constants on the
other side
*Inv. of mult. by 2 is div. by 2
*Visual showing all numbers less than 1/2
on the number line. 
Even though we had a 2 on the right side in line 5, we were dividing
both sides by a positive 2, so we did not change the inequality sign.
Graph:
Since we needed to indicate all values less than 1/2, the part of
the number line that was to the left of 1/2 was darkened.
Since we are not including where it is equal to, an open hole was used. 
Example
11: Solve the inequality and graph the solution set. 
Graph:

*Mult. both sides by LCD
*Get x terms on
one side, constants on the other side
*Inv. of mult. by 1 is div. by 1, so reverse
inequality sign
*Visual showing all numbers less than or equal
to 4 on the number line.

Once again we find ourselves dividing both sides by a negative value,
as shown in line 6. Once we do that, we need to remember to change
the inequality. Note that we still keep the equal part of it.
Graph:
Since we needed to indicate all values less than or equal to 4, the
part of the number line that was to the left of 4 was darkened.
Since we are including where it is equal to, a closed hole was used. 
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 the inequality and graph the solution set.
Need Extra Help on these Topics?
Last revised on July 27, 2011 by Kim Seward.
All contents copyright (C) 2001  2010, WTAMU and Kim Seward. All rights reserved.

