{{ toc.signature }}
{{ toc.name }}
{{ stepNode.name }}
Proceed to next lesson
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}.

# {{ article.displayTitle }}

{{ article.introSlideInfo.summary }}
{{ 'ml-btn-show-less' | message }} {{ 'ml-btn-show-more' | message }} expand_more
##### {{ 'ml-heading-abilities-covered' | message }}
{{ ability.description }}

#### {{ 'ml-heading-lesson-settings' | message }}

{{ 'ml-lesson-show-solutions' | message }}
{{ 'ml-lesson-show-hints' | message }}
 {{ 'ml-lesson-number-slides' | message : article.introSlideInfo.bblockCount}} {{ 'ml-lesson-number-exercises' | message : article.introSlideInfo.exerciseCount}} {{ 'ml-lesson-time-estimation' | message }}
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.

## 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?

## 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

## 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

## 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

## 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

## 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. ## 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.

## 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

## Solution Set of an Inequality

Inequalities compare two quantities and often involve one or more variables. 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
Lastly, the solution set of the inequality can be represented using set-builder notation.
It is worth noting that the solution set of a linear inequality in one variable can also be represented using a number line. ## Theater Auditions

After watching excellent theater performances put on by the drama club, so many students want to join it. Several characteristics are highly sought after for new recruits.

For instance, at the moment, the drama club does not have many tall members. For this recruiting year, they have defined a height to consider for new recruits with the following inequality.
More importantly, the new recruit needs to get a great score in their Literature & Language class. The score is defined with the following inequality.
a Find the solution set of the inequality for the appropriate height of a member.
b Find the solution set of the inequality that represents the score of the literature and language class.

### Hint

a Isolate the variable by using the properties of inequalities.
b Isolate the variable by using the properties of inequalities.

### Solution

a The solution set of an inequality can be found by isolating the variable on one side of the inequality. Simplify the given inequality by using the properties of inequalities to isolate the variable
First, subtract from both sides of the inequality using the Subtraction Property of Inequality.
Next, subtract from both sides of the inequality.
Now, divide both sides of the obtained inequality by to isolate the variable Note that according to the Division Property of Inequality, the inequality sign should be reversed since the divisor is a negative number.
Lastly, rewrite this inequality by using the set-builder notation to represent the answer as a solution set.
This means that this year the drama club is looking for recruits with a height greater than or equal to centimeters.
b Here, the task is to find the solution set of the following inequality.
Once again, proceed in the same way as Part A. Isolate the variable by using the properties of inequalities. That can be done by dividing both sides of the inequality by to get rid of the factor in front of the parentheses.
Next, add  to both sides of the inequality to isolate the variable
Finally, write the simplified inequality in the set-builder notation.
This means that if someone wants to join the drama club, they needs to get a score greater than

## Solving Inequalities

In the applet, there are given various different inequalities. Select the correct solution set for the given inequality. ## Graphing an Inequality on a Number Line

A number line can be used to represent the solution set of an inequality that has one variable. To graph such an inequality, first, determine its type. If it is a strict inequality, then an open boundary point is drawn. Otherwise, a closed boundary point is drawn. Then, the rest of the solution set is shaded accordingly. Consider the following inequality.
The following four steps act as a guide in graphing the given inequality.
1
Determine the Type of Inequality
expand_more
The first step is determining if the inequality is strict or non-strict. In this case, the given inequality is strict because the inequality symbol is
2
Determine the Solution Set and the Boundary Point
expand_more
Next, the solution set and the boundary point of the inequality need to be found. This can be done by solving the inequality using the Properties of Inequalities. Therefore, the boundary point is and the solution set corresponds to all real numbers less than
3
Draw the Boundary Point on the Number Line
expand_more

Here, a circle representing the boundary point is drawn on the number line. If the inequality is strict, the circle is open. If the inequality is non-strict, the circle is closed. For this example, the inequality is strict, and the boundary point is Then, an open circle will be drawn on the number line on the number 4
Shade the Rest of the Solution Set
expand_more

Finally, the rest of the solution set will be shaded by drawing an arrow that goes along the solution set and starts on the boundary point. For this situation, the solution set corresponds to all numbers less than which means the arrow will be along the left of the boundary point. It is worth mentioning that the graph of inequalities whose solution sets are all the real numbers are represented with bidirectional arrows that cover all the number line.

## Showing the Duration of the Plays on a Number Line

The drama club makes a survey of several audiences about the plays presented during the school year. Then they discussed these surveys' results at the end-of-year meeting.

This year all plays were at least minutes long. Since the audiences found the duration of the plays a bit too long according to the survey results, the club takes a decision to keep the next year's plays less than minutes in length.

a Write an inequality for the duration of the next year's plays.
b Choose the graph that describes the solution set of the inequality written in Part A. This will be a way to present the data. ### Hint

a Use a strict inequality to represent the phrase less than.
b The solution set includes real numbers, not only integers.

### Solution

a The club decides to keep the duration of the next year's plays below minutes. Let represent the duration of the plays. Note that less than represent a strict inequality. This means that the inequality sign will be With this in mind, write an inequality to show  is less than
Notice that it can also be written by using the phrasegreater than. In other words, saying is less than  is the same as saying is greater than the duration of the plays .
b Now, the graph of the inequality obtained in Part A can be drawn.
Since the inequality is strict, is not included in the solution set. This means that will be shown with an open circle in the graph. This already eliminates options II and III. Also, notice that the solution set needs to include real numbers, not just integers, because time is a continuous quantity.
The option IV does not make sense for this solution set because it shows only integers. However, Graph I shows exactly the real numbers less than This means that the answer is Graph I.

## Finding the Number of Tickets

At the beginning of the lesson, it was given that the drama club charges per ticket and tickets have been already sold. First, calculate the amount of money earned from these tickets.
The club earns from selling tickets. Recall that the aim is to earn at least from this play. Note that at least can be represented using the inequality symbol less than or equal to This will represent the left hand side of the inequality.
Now, let be the number of additional tickets to be sold after tickets. Then will represent the amount that will be earned. This means that the total amount will be Recall that this amount needs to be greater than or equal to or in other words, needs to be less than or equal to that amount. With this in mind, now complete the other side of the inequality.
Now, solve the obtained inequality to find the minimum number of tickets sold. As before, use the properties of inequalities to isolate the variable Start by subtracting from both sides of the inequality by using the Subtraction Property.
Finally, divide both sides of the inequality by by using the Division Property.
The number of tickets needs to be an integer. The minimum integer value of greater than or equal to is Now, add this number to to find the total number of tickets.
The club needs to sell tickets in total to earn at least

{{ subexercise.title }}