Learning Objectives
Introduction
Tutorial
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
Graphing Inequalities
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
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
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
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.
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).
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
Example
3: Solve the inequality and graph the solution set.
Graph:
*Visual showing all numbers less than 4 on the number line
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.
Graph:
*Visual showing all numbers greater than or = to -5 on the number line.
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
Example
5: Solve the inequality and graph the solution set.
Graph:
*Visual showing all numbers less than -2 on
the number line
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.
Graph:
*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.
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
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:
*Visual showing all numbers less than -14 on
the number line
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.
Graph:
*Visual showing all numbers greater than or
= -3 on the number line
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 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 mult. by -3 is div. both sides by
-3, so reverse inequality sign
*Visual showing all numbers greater than -3
on the number line
Since we are not including where it is equal to, an open hole was used.
Graph:
*Inv. of mult. by 2 is div. by 2
*Visual showing all numbers less than -1/2
on the number line.
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.
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.
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
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?
http://www.sosmath.com/algebra/inequalities/ineq01/ineq01.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.
Last revised on July 27, 2011 by Kim Seward.
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