# Multiplying & Dividing Inequalities

## Definitions

An inequality compares two values.

• Inequality - A comparison of two values or expressions.
For example, 10x < 50 is an inequality whereas x = 5 is an equation.
• Equation - A statement declaring the equality of two expressions.
For example, 4x = 8 is an equation whereas 10x > 20 is an inequality.

## Operating With Inequalities: Multiplying & Dividing

Performing multiplication or division with an inequality is nearly identical to multiplying or dividing parts of traditional equations (with one exception, covered below).

Consider the following examples:

10x + 15 < 25 + 5x
10x + 15 - 15 < 25 - 15 + 5x
10x < 10 + 5x
10x - 5x < 10 + 5x - 5x
5x < 10
x < 2 ### The Exception: Negative Numbers

There is one very important exception to the rule that multiplying or dividing an inequality is the same as multiplying or dividing an equation.

Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.

In the following example, notice how the < sign becomes a > sign when the inequality is divided by -2

-2x - 10 < 2
-2x - 10 + 10 < 2 + 10
-2x < 12
x > -6 [Dividing by -2 required the flipping of the inequality sign]

In the following example, notice how the < sign becomes a > sign when the inequality is divided by -2

-2x + 15 < 3
-2x + 15 - 15 < 3 - 15
-2x < -12
x > 6 [Dividing by -2 required the flipping of the inequality sign]

#### Warning: Caution When Multiplying or Dividing Variables

One very important implication of this rule is: You cannot divide by an unknown (i.e., a variable) unless you are sure of its sign since you do not know whether you must flip the sign of the inequality. There are plenty of instances where you will know the sign of a variable and as a result, you can multiply or divide and know for sure whether you must flip the inequality sign. However, you must always ask yourself whether you know for sure the sign of the variable before dividing or multiplying when dealing with an inequality.

If 2x5y < 10y, what is the range of potential values for x?
You cannot divide by y or 5y since you do not know whether y is negative or positive and, as such, you do not know whether to flip the inequality.

## 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 using multiplication or division, the most important part is to solve each inequality separately and then combine them.

If 2x < 10, -5x < -10, and 15x < 150, what is the range of possible values for x?

1.) Solve each inequality alone.
2x < 10
x < 5 [Note: The inequality is not flipped since we are dividing by 2, which is positive]

-5x < -10
x > 2 [Since we divide by -5, a negative number, we flip the inequality sign]

15x < 150
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 > 2
x < 10

The area of overlap--i.e., the solution to the set of inequalities--is where x < 5 and x > 2

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. 