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Inequalities can be solved like equations. The key step is to isolate the variable on one side of the inequality. Inequalities may have multiple, or even infinitely many, solutions. In this lesson several examples are introduced showing how to use the properties of inequalities to solve them.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

## Tickets for a Drama: Making Money

Jordan is a member of the drama club at her school. The club plans to stage the play The Little Frog in Town this weekend. They charge per ticket.

a The club hopes to earn at least from the play. If the club has already sold tickets, write an inequality to represent the number of tickets they still need to sell.
b What is the minimum total number of tickets the club needs to sell to reach their goal?
Discussion

## Properties of Inequalities

Not all inequalities are expressed in the form or Yet, through inverse operations and the Properties of Inequalities, any inequality can be simplified to one of the mentioned forms. Consider the Addition and Subtraction Properties of Inequalities.

Rule

Adding the same number to both sides of an inequality generates an equivalent inequality. This equivalent inequality will have the same solution set and the inequality sign remains the same. Let and be real numbers such that Then, the following conditional statement holds true.

If then

This property holds for the other types of inequalities.

### Proof

The case when will be proven. The remaining cases can be proven similarly. Before starting the proof, the following biconditional statement needs to be considered.
Now, the Identity Property of Addition can be applied to the second part of the statement.

Rewrite as

Using the biconditional statement, the last inequality can be rewritten.
Finally, because the property is obtained.

If then

Discussion

## Subtraction Property of Inequality

Subtracting the same number from both sides of an inequality produces an equivalent inequality. The solution set and inequality sign of this equivalent inequality does not change. Let and be real numbers such that Then, the following conditional statement holds true.

If then

This property holds for the other types of inequalities.

### Proof

Subtraction Property of Inequality
The case when will be proven. The other cases can be proven using a similar reasoning. Consider the biconditional statement before beginning the proof.
This property can be proven using the Additive Inverse of which is Now, the Identity Property of Addition can be applied to the second part of the statement.

Rewrite as

From the biconditional statement, the last inequality can be rewritten.
Finally, because the property has been proven.

If then

Discussion

## Multiplication Property of Inequality

Multiplying both sides of an inequality by a nonzero real number produces an equivalent inequality. The following conditions about need to be considered when applying this property.

 Positive If is positive, the inequality sign remains the same. If is negative, the inequality sign needs to be reversed to produce an equivalent inequality.

For example, let and be real numbers such that and Then, the equivalent inequalities can be written depending on the sign of

• If and then
• If and then
This property holds for the other types of inequalities.

### Proof

Multiplication Property of Inequality

The case when will be proven. The remaining cases can be proven following a similar reasoning. Before starting the proof, the following properties of real numbers need to be considered.

• if and only if
• If and are positive, then
• If is negative, then is positive.

Using these properties, the following conditional statements can be proven.

• If and then
• If and then

Each conditional statement will be analyzed separately.

### When Is Greater Than

It is given that then using the first property, it is known that is greater than
Furthermore, because from the second property, it can be stated that the product of and is also greater than
Now, the second part of this conditional statement can be rewritten using the Distributive Property.
From the first property, it can be said that if and only if Additionally, because the conditional statement has been proven.
If and then

### When Is Less Than

Again, because the following statement is valid.
Additionally, since from the third property it follows that is positive. Moreover, the product of and will be positive.
Now, can be distributed in the second part of the statement.
Simplify
Finally, because the property has been proven.

If and then

Discussion

## Division Property of Inequality

Dividing both sides of an inequality by a nonzero real number produces an equivalent inequality. However, the following conditions need to be considered.

 Positive If is positive, the inequality sign remains the same. If is negative, the inequality sign needs to be reversed to produce an equivalent inequality.

For example, let and be real numbers such that and Then, the equivalent inequalities can be written depending on the sign of

• If and then
• If and then
This property holds for the other types of inequalities.

### Proof

Division Property of Inequality

The case when will be proven. The remaining cases can be proven following a similar reasoning. Before starting the proof, the following properties of real numbers need to be considered.

• if and only if is positive.
• If and are positive, then is also positive.
• If is negative, then is positive.

Using these properties, the following conditional statements can be proven.

• If and then
• If and then

Each case will be analyzed separately.

It is given that then using the first property, it is known that is greater than
Furthermore, because from the second property, it can be stated that divided by is also greater than
Now, the second part of this conditional statement can be rewritten.
By using the first property, it can be said that is less than Additionally, because the property has been proven.

If and then

Again, because the following statement is valid.
Additionally, since from the third property, it follows that is positive. Moreover, the quotient of and will be positive.
Now, the second part of this statement can be rewritten.
Simplify
Finally, because the property has been obtained.

If and then

Pop Quiz

## Identifying the Appropriate Property to Solve the Inequality

In the applet, determine the property used to isolate the variable on one side of the given inequality as shown.

Discussion

## Equivalent Inequalities

Two or more inequalities are equivalent inequalities if they have the same solution set. Similar to equivalent equations, applying Properties of Inequality to an inequality produces an equivalent inequality. Consider the following example.
Three Properties of Inequalities will be used to solve this inequality. The solution set of every inequality created in this process is The Properties of Inequalities produced three equivalent inequalities until the solution set was found.
Example

## Number of Seats in the Theater Hall

The play The Little Frog in Town received rave reviews from the audience after its first showing. The club decides to put on this play again with one major difference — they will hold it in the greatest theater hall their city has to offer! Now, the drama club needs to know the number of seats in the hall to print new tickets.

The hall consists of two floors, the ground floor and the balcony. The number of seats on the ground floor is defined by the following inequality.
The number of seats in the balcony is defined by another inequality.
a Find the simplest form of the inequality for the number of ground floor seats.
b Find the simplest form of the inequality for the number of seats on the balcony.
c What is the maximum number of seats in the theater hall according to the given inequalities?

### Hint

a Use the properties of inequalities to simplify the given inequality.
b Use the properties of inequalities to simplify the given inequality.
c Use the simplified inequalities from Part A and Part B.

### Solution

a Start by examining the given inequality that represents the number of seats on the ground floor.
Now, use the properties of inequalities to simplify this inequality. Notice that the variable appears on both sides of the inequality. First, subtract from both sides of the inequality by using the Subtraction Property of Inequality.
Next, add to both sides of the inequality by using the Addition Property of Inequality.
Finally, divide both sides of the inequality by to get the variable alone by using the Division Property of Inequality.
The simplest form of the given inequality is obtained.
This means that the number of seats on the ground floor is less than or equal to In other words, there are at most seats on the ground floor.
b This time, the following inequality will be simplified.
Once again, use the properties of inequalities to simplify it. Start by adding to both sides of the inequality by using the Addition Property of Inequality.
Now, add  to both sides of the inequality.
As the last step, divide both sides of the inequality by Note that since is a positive number, the inequality sign stays the same. In the case that the divisor is a negative number, the inequality sign should be reversed.
The simplest form of the given inequality is found.
This means that is greater than the number of seats on the balcony. In other words there are at most seats on the balcony.
c Recall the simplified forms of the inequalities in Part A and Part B.
According to these inequalities, there are at most seats on the ground floor and seats on the balcony. Now, add these number of seats to find the maximum number of seats in the theater hall.
The maximum number of seats in the theater hall is
Discussion

## Solution Set of an Inequality

A solution of an inequality is any value of the variable that makes the inequality true. As an example, consider the following inequality.
Notice that if is substituted for in the inequality, the inequality holds true. Therefore, it can be said that is a solution to the given inequality.
However, this is not the only value that makes the inequality true. There are other values like and that make it true. The set of all possible values that satisfy an inequality is the solution set of an inequality. The solution set can be determined by applying the Properties of Inequalities to isolate the variable on one side of the inequality.
Solve for