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Many situations involve finding the optimal amount of time, money, or a certain quantity of material to fulfill the requirements of some task that could have more than one solution. Some of these situations can be expressed as inequalities. This lesson will explore how to model specific situations as linear inequalities involving one variable and represent their solution set in a number line.

Catch-Up and Review

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

Challenge

How Can Real-Life Situations Be Modeled to Help in Making Choices?

Ignacio, has just been offered a paid internship as a junior rocket scientist for a space exploration company. The company has offered Ignacio two options for how to be paid.

Salary options. The first option is $3500 plus 12% of sales. The secon option is $5000 plus 5% of sales

Ignacio is unsure which is the best option. Help him make the best choice by answering the following questions. His salary depends on it!

a Write an inequality representing the earnings from his sales that guarantees option is better than option for Ignacio.
b If Ignacio is sure that he will make at least worth of sales per month, then which option should he choose?
c Graph the solution set of the inequality found in Part A on a number line.

Explore

Inequalities Come In Different Forms

Inequalities have many forms. Some have the variable on one side and others have them on both sides of the inequality. They can also contain constant terms that are being subtracted from or added to the variable. Examples of these situations can be seen in the following applet.
A set of inequalities
Think about the following questions.
  1. What happens to an inequality if one number is added to both sides?
  2. What if one number is subtracted from both sides?
  3. How does performing these operations affect the inequality sign?
  4. What happens to the solution set of the inequality?

Discussion

Addition and Subtraction Properties of Inequalities

Similar to equations, inequalities have some properties that allow their manipulation without changing their solution set. When these properties are applied, an equivalent inequality is obtained. The Addition and Subtraction Properties are two of them.

Rule

Addition Property of Inequality

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.
The Addition Property of Inequality for All Types of Inequalities

Proof

Addition Property of Inequality
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

Rule

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.
The Subtraction Property of Inequality for All 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

Explore

Inequalities Involving a Variable With a Real Number Coefficient

The Addition and Subtraction Properties of Inequalities can help to isolate the variable on one side of the inequality by creating equivalent inequalities. Although some inequalities can be solved by using these two properties, there are inequalities where the other properties of inequalities need to be used to determine the solution set.
A set of inequalities with the variable term on one side of the inequality
As can be seen, when the variable in an inequality is multiplied by a real number it cannot be isolated on one side of the inequality by using the aforementioned properties. Now, think about the following questions.
  1. What happens to an inequality if both sides are multiplied by the same positive number?
  2. What if both sides are multiplied by a negative number instead?
  3. What if both sides of the inequality are divided by the same positive number?
  4. What if both sides are divided by a negative number?

Discussion

Multiplication and Division Properties of Inequalities

The Addition and Subtraction Properties of Inequalities do just a portion of the work. That is because they do not create the ability to isolate variable terms that contain coefficients. Not to worry, the Multiplication and Division Properties can help in these cases. Together, these properties help to solve inequalities by creating equivalent inequalities.

Rule

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.
Negative 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.
The Multiplication Property of Inequality for All 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

Rule

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.
Negative 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.
The Division Property of Inequality for All the 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 Some Properties of Inequalities

Knowing which property to use when solving an inequality is of importance because this can minimize mistakes. In the applet, select the property used to produce each equivalent inequality.

An applet showing different inequalities and its equivalent inequality that results of applying one of the properties of inequalities

Discussion

Determining and Representing the Solution of Inequalities

Applying the Properties of Inequalities to one inequality will produce equivalent inequalities. These equivalent inequalities can have a simpler form, making their solutions more straightforward to identify. Since the equivalent inequalities have the same solutions, the solution set of the original inequality can be determined.

Concept

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.

Method

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
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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
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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
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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

The boundary point on a number line


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.

The boundary point on a number line

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.

Example

Using Inequalities to Help a Company Explore Outer Space

An object must travel at a speed of at least kilometers per second to escape Earth's gravitational field. At Gravitasi Z, engineers built a rocket and were tasked with the mission of exploring celestial objects faraway from Earth. However, there is a big problem: The rocket was made to travel at a speed of only kilometers per second!

A rocket taking off from the earth

Gravitasi Z's engineers plan to improve the rocket in order to accomplish its mission. Solve the following predicaments to help them succeed.

a They need an inequality expressing the speed that should be added to the rocket to surpass the Earth's gravitational field. Write this inequality.
b The engineers wonder by how much, at the least, should the rocket's speed be increased to overcome the gravitational force? Help them find this speed.
c A few junior engineers are staring at some graphs, not quite sure of their meaning.
Number lines representing the possible solution sets of the inequality
They need to know which of the graphs describes the solution set of the inequality. Let them know which is the correct graph.

Hint

a Begin by writing an expression for the final speed of the rocket.
b Solve the inequality found in Part A. The boundary point represents the minimum speed that is missing.
c Draw the boundary point on a number line. Then, identify which side of the boundary point represents the solution set.

Solution

a It is known that the rocket can reach a speed of kilometers per second. Let represent the additional speed that the rocket will gain after improvements. Then, the sum of and will be the speed of the rocket after the improvements.
The company needs the final speed to be at least kilometers per second. The phrase at least means greater than or equal to. Therefore, the inequality is non-strict and the symbol must be
b It is asked to find the minimum speed that should be added to the rocket, the minimum value of To do so, the inequality written in Part A should be solved. Using the Subtraction Property of Inequality, can be isolated.
The boundary point is representing the minimum speed that needs to be added to the rocket. This inequality also means that improvements that cause the rocket to speed up greater than or equal to kilometers per second will enable the rocket to start its mission.
c The graph of the inequality written in Part A should be determined.
To graph this inequality on a number line, the first step is to determine its type. It is a non-strict inequality because it involves the symbol Its solution set and boundary point were found in Part B.
Solution Set Boundary Point

Since the inequality is non-strict, a closed circle will be drawn on the number line on its boundary point

A number line with the boundary point of the inequality

Now, the rest of the solution set needs to be shaded. Because the speed added needs to be greater than or equal to the region on the right of the boundary point will be shaded.

The solution set of the inequality on a number line

This corresponds to Graph II.

Example

Using Inequalities to Understand Profits

Ignacio is an employee at Gravitasi Z. He feels unsure about the success of their space exploration, so he decides to diversify his money making opportunities by investing in cryptocurrency. He has found a crypto exchange company that does all of the actual investing work for him.

A bag of money

The exchange company charges a subscription of plus a commission of for every cryptocurrency bought. Also, for an additional fee of each cryptocurrency can be sold again later. Ignacio wants to determine the number of cryptocurrencies that will make this investment profitable. Fractions of cryptocurrencies can be bought.

a Write an inequality that expresses the number of cryptocurrencies Ignacios needs to buy to make a profit.
b What is the minimum number of cryptocurrencies that Ignacio needs to buy to at least make some profit? Give the minimum number in integer form.
c The investment company presents the following graphs to Ignacio.
Number lines representing the possible solution sets of the inequality
Which of the graphs describes the solution set of the inequality?

Hint

a Begin by writing an expression for the total investment. Then, find an expression for the total sales expected after selling all the cryptocurrencies.
b Solve the inequality found in Part A using the Properties of Inequalities.
c Draw the boundary point on a number line. Then, identify which side of the boundary point represents the solution set.

Solution

a Let be the number of cryptocurrencies. Because the company charges a subscription of plus per cryptocurrency bought, an expression for the total investment can be written by adding the product of and to the subscription cost.
Additionally, it is given that each cryptocurrency can be later sold for each. Therefore, multiplying and will give the total amount recovered after selling all the cryptocurrencies.
Finally, to make a profit, the total sales need to be greater than the total investment. That means the inequality is strict and its symbol is The inequality representing the number of cryptocurrencies needed to buy to make a profit can be written using this information.
b To help Ignacio find the minimum number required to expect make some profit, the inequality needs to be solved. To do so, the Properties of Inequalities can be used. In this case, the Subtraction Property of Inequality should be used first.
The boundary point is about This means that as long as Ignacio bought more than cryptocurrencies, he can expect to make a profit. Therefore, the minimum he needs to buy is
c Consider the inequality found in Part A.
To graph the solution set of this inequality, the first step is determining its type. Because the inequality symbol is it is a strict inequality. The solution set and the boundary point of this inequality were found in the previous part.
Solution Set Boundary Point
Since the inequality is strict, an open circle will be drawn on the number line on its boundary point .
A number line with the boundary point of the inequality

Now, the rest of the solution set needs to be shaded. Because the number of cryptocurrencies bought needs to be greater than the region on the right side of the boundary point will be shaded.

The solution set of the inequality on a number line

This corresponds Graph I.

Example

Inequalities With Only Integer Solutions

Gravitasi Z wants to sale coffee mugs to the families of their astronauts — at a low price, of course. A company that produces mugs wants to charge Gravitasi Z a fixed amount of for production plus a tax of per mug. Gravitasi Z plans to sell each mug for

a cup of coffee with a personalized logo

The space exploration company, busy with making rockets, decides to hire a financial advisor to make these calculations. They hope to make at least a small profit. Find the answers to the following situations to become Gravitasi Z's trusted financial advisor.

a Write an inequality representing the number of mugs that Gravitasi Z should order that will guarantee they make a profit. Assume that they will sell all of the mugs that are ordered to be produced.
b Gravitasi Z gave a small budget to this project, so they want to invest in the minimum number of mugs that guarantee a profit. What is this minimum number?
c Choose the graph that describes the solution set of the inequality written in Part A. This will be a way to present the data.
Number lines representing the possible solution sets of the inequality

Hint

a Begin by writing an expression for the total expenses and the total sales.
b Can a fraction of a mug be sold?
c The graph of an inequality with integer solutions is represented by plotting points on the values representing its solution set.

Solution

a Let be the number of mugs that they will order from the production company. The cost of production is plus a tax of per mug. Then, the total expenses can be expressed as the sum of and the product of and
Since each coffee mug can be sold for assuming that all mugs will be sold, the total sales can be expressed by the product of and the number of mugs
In order for Gravitasi Z to profit from this project, the total expenses should be less than the total sales. Therefore, the relationship between these two quantities can be shown by the inequality symbol
b Gravitasi Z wants the minimum number of coffee mugs that guarantee that they will make a profit. To do so, the inequality written in the previous part will be solved using the Properties of Inequalities.
Solve for
From this inequality, it can be seen that is the boundary point. Recall that represents the number of mugs. Therefore, must be a positive integer. Combining these findings, the solution set of the inequality can be written as follows.
The minimum number of mugs that Gravitasi Z should order from the production company is
c To find the answer, the solution set of the inequality will be graphed. The solution set and the boundary point of the inequality were found in the previous part.
Solution Set Boundary Point

Since the inequality is