Inequalities
Inequalities are a way to express the relative size of two numbers or objects, that being that one of those objects is smaller, greater or equal in size in relation to the other object. You can express inequalities graphically via a number line or as a mathematical expression. A number line is a straight line on which every point represents a single real number and it is very easy to portrait inequalities on it. For example, let us say you get an expression that says:
X is greater than 9.
The solutions of this problem (or the value of X) are values on the number line that are greater than 9 and you should mark these values. Be careful not to mark the number 9 as well, because the assignment asks only for the values greater than that number. Now, let us say that the assignment states that:
X is greater than or equal to 9
In this case the solution would include the number 9 because it is permitted in the text of the assignment that x can be equal to 9.
Inequalities also come in forms of longer mathematical expressions, similar to equations. Depending on the number of steps you have to perform in order to solve them, they can also be divided into onestep inequalities, twostep inequalities or multistep inequalities.
Inequalities can be in five different ways and each of them represents different relations between the objects that are being compared. We are going to explain them to you in this table using both mathematical symbols and text.
Symbols 
Meaning 
a < b 
The value of a is less than the value of b 
a > b 
The value of a is greater than the value of b 
a ≠ b 
The value of a is not equal to the value of b 
a ≤ b 
The value of a is less than or equal to the value of b 
a ≥ b 
The value of a is greater than or equal to the value of b 
The expression for the value of a being not equal to the value of b can also be expressed with the symbol “<>” instead of the symbol “≠”, but they mean the same. In this case, a can be any number on the number line except b.
Properties of inequalities
Inequalities have five basic properties: transitivity, addition and subtraction, multiplication and division, additive inverse and finally, multiplicative inverse.
The property of transitivity states that for a, b, and c being any real numbers:
If 
a < b and b > c 
then 
a > c 
If 
a > b and b < c 
then 
a < c 
If 
a ≤ b and b = c 
then 
a > c 
If 
a ≥ b and b = c 
then 
a < c 
The rules that deal with the properties of addition and subtraction within inequalities say that for a, b and c being any real numbers:
If 
a < b 
then 
a + c < b + c 
and 
a – c < b – c 
If 
a > b 
then 
a + c > b + c 
and 
a – c > b – c 
The rules that deal with the properties of multiplication and division within inequalities state that for a, b and c being any real numbers:
If 
c positive and a < b 
then 
ac < bc 
and 
a/c < b/c 
If 
c negative and a < b 
then 
ac > bc 
and 
a/c > b/c 
The property of the additive inverse says that for a and b being any real numbers:
If 
a < b 
then 
 a > – b 
If 
a > b 
then 
 a < – b 
The property of the multiplicative inverse states has two parts. If a and b are both positive or both negative real numbers, then:
If 
a < b 
then 
1/a > 1/b 
If 
a > b 
then 
1/a < 1/b 
If either a or b is negative, then:
If 
a < b 
then 
1/a < 1/b 
If 
a > b 
then 
1/a > 1/b 
If you wish to practice with inequalities, check the free worksheets below.
Inequalities exams for teachers
Exam Name  File Size  Downloads  Upload date 
Integers


Graph inequalities – Integers – easy  245.3 kB  3489  October 13, 2012 
Graph inequalities – Integers – medium  234.6 kB  2682  October 13, 2012 
Graph inequalities – Integers – hard  238.7 kB  2025  October 13, 2012 
Decimals


Graph inequalities – Decimals – easy  223.2 kB  1119  October 13, 2012 
Graph inequalities – Decimals – medium  221 kB  1133  October 13, 2012 
Graph inequalities – Decimals – hard  224.6 kB  945  October 13, 2012 
Fractions


Graph inequalities – Fractions – easy  232.4 kB  796  October 13, 2012 
Graph inequalities – Fractions – medium  243.1 kB  640  October 13, 2012 
Graph inequalities – Fractions – hard  270 kB  655  October 13, 2012 
Inequalities worksheets for students
Worksheet Name  File Size  Downloads  Upload date 
Graph inequalities – Integers  2.4 MB  3120  October 14, 2012 
Graph inequalities – Decimals  2.2 MB  1048  October 14, 2012 
Graph inequalities – Fractions  746.5 kB  1128  October 14, 2012 
Inequality for given graph  527.1 kB  1856  October 14, 2012 