## Definitions

An inequality compares two values.

• Inequality - A comparison of two values or expressions.
For example, x < 3 is an inequality whereas x = 3 is an equation.
• Equation - A statement declaring the equality of two expressions.
For example, 2x + 7 = 17 is an equation whereas 2x + 7 > 17 is an inequality.

## Operating With Inequalities: Adding & Subtracting

Operating with inequalities is virtually the same as operating with traditional equations (Note: One big difference does exist and is covered in the multiplying and dividing inequalities section). Just as you can add or subtract numbers in traditional equations, so you can add or subtract numbers to and from inequalities. Likewise, the same process used to solve for an unknown in an equation is also used in solving for an unknown within an inequality.

Consider the following examples:

x + 7 > 17
x + 7 - 7 > 17 - 7
x > 10

15x + 15 < 25 + 5x
15x - 5x + 15 < 25 + 5x - 5x
10x + 15 < 25
10x + 15 - 15 < 25 - 15
10x < 10
x < 1

10x2 + 5x + 17 > 10(x2 + x) + 2
10x2 + 5x + 17 > 10x2 + 10x + 2
10x2 + 5x + 17 -2 > 10x2 + 10x + 2 - 2
10x2 + 5x + 15 > 10x2 + 10x
10x2 - 10x2 + 5x + 15 > 10x2 - 10x2 + 10x
5x + 15 > 10x
5x - 5x + 15 > 10x - 5x
15 > 5x
3 > x
x < 3

## Multiple Inequalities

Just as it is possible to solve two simultaneous equations, so it is possible to solve two inequalities (or three, or four, etc.). In solving multiple simultaneous inequalities, the most important part is to solve each inequality separately and then combine them.

If x + 10 > 15, x + 7 > 14, and 10x < 100, what are the possible values of x?

1.) Solve each inequality alone.
x + 10 > 15
x + 10 - 10 > 15 - 10
x > 5

x + 7 > 14
x + 7 - 7 > 14 - 7
x > 7

10x < 100
x < 10

2.) Combine each inequality and find the overlap (i.e., the areas where each inequality is satisfied--this area is the solution).
x > 5
x > 7
x < 10

The area of overlap--i.e., the solution to the set of inequalities--is where x > 7 and x < 10

For many students, the above set of inequalities can best be understood graphically. The solution to the set of inequalities is the overlapping graphical area. 