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10. Using Inductive Reasoning to Create Inequalities
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10. 

Using Inductive Reasoning to Create Inequalities

Inductive reasoning is a method used to form inequalities, which are mathematical expressions that compare two quantities. The lesson focuses on two types of inequalities: strict and non-strict. Strict inequalities do not include the boundary values in the solution set, while non-strict inequalities do. These types of inequalities have practical applications in real-life situations. For example, they can be used to model the height of basketball players or the volume of air left in a swimmer's lungs. Understanding the nuances between strict and non-strict inequalities can help in making more accurate models for problem-solving in fields like sports, healthcare, and finance.
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9 Theory slides
9 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Using Inductive Reasoning to Create Inequalities
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One reason for learning mathematics could be to model real-life situations. With those models, the situations can be interpreted and some inferences can be made. This lesson will focus on creating models for real-life situations that are restricted by some rules.

Catch-Up and Review

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

Challenge

Matching the Key Phrases of Inequalities

Ramsha is given the following table on a math test. There are four different inequalities, and for each inequality there are several key phrases.
She is asked to fill in the table. Help her by moving the key phrases to the corresponding columns.
Explore

Examining the Key Phrases of Inequalities

A delivery company is trying to organize its shipments. Help them by loading the weight required to satisfy the given statement onto the truck.
What can be concluded about phrases such as at most, less than, and at least that restrict the amount of the weight?
Discussion

Inequality

An inequality, like an equation, is a mathematical statement that compares two quantities. An inequality contains the symbols <, >, ≤, or ≥. There are several ways each inequality can be phrased.

Inequality Symbol Key Phrases
< & ∙ is less than & ∙ is fewer than
> & ∙ is greater than & ∙ is more than
≤ & ∙ is less than or equal to & ∙ is at most & ∙ is no more than
≥ & ∙ is greater than or equal to & ∙ is at least & ∙ is no less than

With an inequality, it is possible to compare any combination of two numbers, variables, numeric expressions, or algebraic expressions.

Symbol Example Meaning
< x<1 The variable x is less than 1.
≤ x+1 ≤ -3 The algebraic expression x+1 is less than or equal to - 3.
> 2x-5 > 5 The expression 2x-5 is greater than 5.
≥ x ≥ 2x+1 The variable x is greater than or equal to the expression 2x+1.
As shown in the table, some inequalities indicate that the two quantities are not necessarily equal, whereas others indicate that they are strictly never equal.
Concept

Strict vs. Non-Strict Inequality

An inequality that compares two quantities that are strictly not equal is called a strict inequality. There are two types of strict inequalities. Less Than:&< Greater Than:&> The boundary values in strict inequalities are not included in the solution set. On the other hand, an inequality that compares two quantities that are not necessarily different is called a non-strict inequality. There are two types of non-strict inequalities. Less Than or Equal To:&≤ Greater Than or Equal To:&≥ The boundary values in non-strict inequalities are included in the solution set. Consider the graphs of several examples of strict and non-strict inequalities.
Strict vs. Non-strict Inequalities shown in the number line and in the cartesian plane
It can be seen that in order to indicate strict inequalities graphically, an open point (∘) is used for number line inequalities and a dashed boundary line or curve is used for two-dimensional inequalities. To indicate non-strict inequalities graphically, a closed point (∙) is used for number line inequalities and a solid boundary line or curve is used for two-dimensional inequalities.
Example

Modeling the Volume of Air Left Using an Inequality

It is the weekend and Tadeo is at home, flipping through channels on the TV. He decides to make up math questions about whatever is on each channel to entertain himself. On the sports channel is a program about Olympic swimmer Michael Phelps. The program claims that Phelps has a lung capacity of 12 liters, nearly twice that of an average adult.

Michael Phelps.jpg

If Phelps's lungs are full and he exhales at a rate of 1.2 liters per second in a swimming race, help Tadeo write an inequality that models when Phelps will still have more than 5.1 liters of air left in his lungs.

Hint

Let t be the variable representing the number of seconds Phelps exhales air from his lungs.

Solution

In this context, the unknown quantity is time. Let t be the variable representing the number of seconds Phelps exhales air from his lungs. From here, an expression for the volume of the air exhaled after t seconds can be written by multiplying the rate 1.2 liters per second and t. Volume of Exhaled Air 1.2 t From the given information, Phelp's lung capacity is 12 liters. By subtracting the volume of the exhaled air from the total lung capacity, the volume of the air left in his lungs can be found. Volume of Air Left 12- 1.2 t Now that the expression for the volume of the air left in his lungs has been written, next the inequality symbol needs to be determined. Since the volume of the air left is more than 5.1 liters, the inequality is strict and the symbol > should be used. 12- 1.2 t > 5.1

Example

Modeling the Trunks of Trees Using an Inequality

The nature channel is showing a program produced by the National Association of State Forestry about shade trees. Tadeo learns that both the cottonwood and the red maple are shade trees and that the diameter of a red maple trunk grows by about 0.31 inches per year, while the diameter of a cottonwood trunk grows by about 0.64 inches per year.

Red Maple and Cottonwood

Assume that a red maple with a trunk diameter of 3.4 inches has been planted together with a cottonwood with a trunk diameter of 2.7 inches. Help Tadeo write an inequality that models when the red maple will be thinner than the cottonwood.

Hint

Let t be the variable representing the number of years that pass.

Solution

In this context, the unknown quantity is time. Let t be the variable representing the number of years that pass. From here, an expression for the change in diameter of each tree can be written by multiplying t by the growth rate. Red Maple & Cottonwood 0.31 t & 0.64 t It is given that at the start of the time period, the trunk diameter of the cottonwood is 2.7 inches and the trunk diameter of the red maple is 3.4 inches. By adding the changes in diameters to the starting diameters, the trunk diameters of the trees after t years can be calculated. Red Maple & Cottonwood 3.4+ 0.31 t & 2.7+ 0.64 t Now that each tree has an expression for its diameter, next the inequality symbol needs to be determined. Note that the trunk diameter of the red maple should be thinner than the that of the cottonwood. Therefore, the inequality will be strict and the < symbol should be used. 3.4+ 0.31 t < 2.7+ 0.64 t

Example

Modeling an Savings Using an Inequality

The next channel is showing reruns of one of Tadeo's favorite TV shows, Supernatural. The show features a 1967 Chevrolet Impala. Tadeo decides to save up for a car just like this one.

1967 Impala.jpg

After some research online, Tadeo learns that a 1967 Chevrolet Impala is available in his town for about $23 000. Currently, he has $500 in savings. He plans to save about $200 each month until he can afford the car. He wants to save at least $25 000 in case of any price changes before he can afford it. Write an inequality that models when Tadeo will have saved at least $25 000.

Hint

Let m be the variable representing the number of months Tadeo saves money.

Solution

In this context, the unknown quantity is time. Let m be the variable representing the number of months Tadeo saves money. With this variable, an expression can be written for the amount of money Tadeo can save. Recall that he plans to save $200 each month. Money From the Monthly Savings 200 m Since he starts with $500, his total savings after m months can be represented by the following expression. Total Savings 500+ 200 m To write the inequality, the next step will be determining the inequality sign. Remember that he will need at least $25 000. This means that he will need either $25 000 or more than $25 000. Since the possibility of an equality exists, the inequality will be non-strict and the symbol ≥ should be used. 500+ 200 m ≥ 25 000

Example

Modeling Trips to the Amusement Park Using an Inequality

Tadeo sees a segment for a local amusement park on the news. According to the segment, an annual pass to the park is $129.99. Tadeo knows that round-trip bus fare to the amusement park is $10 per trip.

Amusement Park
Tadeo wants to use some of his savings buy a pass. He hopes to visit the amusement park regularly throughout the year with his friends. Supposing he uses no more than $500 of his savings, write an inequality representing the number of times Tadeo can visit the park if he also has to purchase the annual pass.

Hint

Let t be the variable representing the number of round trips Tadeo can take on the bus.

Solution

In this context, the unknown quantity is how many times Tadeo can visit the park, or the number of trips he can take on the bus. Let t be the variable representing the number of trips he can take. From here, an expression for the cost for each round-trip charge can be written as follows. Cost For Round-Trip Bus Fare 10 t Since the initial cost of the annual park pass is $129.99, the total cost after t trips can be expressed by the sum of the initial fee and cost of bus fare per trip. Total Cost 129.99+ 10 t Remember that Tadeo wants to spend no more than $500. Therefore, the total cost must be less than or equal to $500. This means that the inequality will be non-strict and the ≤ symbol should be used. 129.99+ 10 t ≤ 500

Closure

Properties of Inequality

Several word problems have been modeled by different inequalities throughout this lesson. However, before ending the lesson, there are some properties should be mentioned. Similar to the properties of equality, inequalities also have some properties. Note that even if these properties are closely related to the properties of equality, there are important differences.

Rule

Anti Reflexive Property of Inequality

A real number can never be less than or greater than itself.

x ≮ x and x ≯ x

This property is an axiom. Therefore, it can be accepted as true without proof.
Rule

Anti Symmetric Property of Inequality

For any two real numbers x and y, if x is less than y, then y cannot be less than x.

If x< y, then y ≠< x.

Alternatively, if x is greater than y, then y cannot be greater than x.

If x> y, then y ≠> x.

This property is an axiom. Therefore, it can be accepted as true without proof.
Rule

Transitive Property of Inequality

Let x, y, and z be real numbers. If x is less than y and y is less than z, then x is less than z.

If x< y and y< z, then x < z.

This property also applies to other types of inequalities — >, ≤, and ≥.

  • If x> y and y> z, then x > z.
  • If x≤ y and y≤ z, then x ≤ z.
  • If x≥ y and y≥ z, then x ≥ z.
Since this property is an axiom, it does not need proof to be accepted as true.


Level 1
Level 2
Level 3
Using Inductive Reasoning to Create Inequalities
Exercise 1.1

An inequality can contain the symbol <, >, ≤, or ≥. For each of these symbols, there are several key phrases that represent them. Here, one key phrase is given per inequality symbol.

Inequality Symbols and Key Phrases
Match these key phrases with their corresponding symbols.

Let's begin by recalling the basic key phrases for the inequality symbols.

Basic Key Phrase Inequality Symbol
is less than <
is greater than >
is less than or equal to ≤
is greater than or equal to ≥

Looking at this table, we can immediately match the key phrase is greater than with its inequality symbol, >. Next, by considering the synonyms of the other key phrases we can match them with their inequality symbols. Basic Key Phrase& Synonym is less than & is fewer than is less than or equal to & is at most is greater than or equal to & is no less than From here, each key phrase can be matched with its corresponding symbol.

Key Phrase Inequality Symbol
is fewer than <
is greater than >
is at most ≤
is no less than ≥
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Hint & Answer
S
Solution

What inequality represents the verbal expression?

A number t plus 5 is less than 11.

7 is more than twice a number k minus 1.

To write a verbal expression as an algebraic expression, we will start by identifying the inequality symbol we need to use and placing it in the center of our statement. In this case, we have the key phrase is less than, which is represented by the symbol <.

On the left-hand side of the inequality we have a number t plus 5, which can be expressed as t+5.

On the right-hand side of the inequality, we have 11. We can now form our final answer.


By following the same procedure as in Part A, we will first determine the inequality symbol. The key phrase for the inequality is is more than, which can be represented by >.

This time, we have a numeric expression on the left-hand side that is 7.

On the right-hand side, twice a number k minus 1 can be expressed as 2k-1. With this information, we can finally complete our inequality.


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Hint & Answer
S
Solution

What inequality represents the verbal expression?

Four times a number m minus 2 is no more than the quotient of a number n and 5.

The quotient of 4 and a number s is at least the difference of a number d and 13.

To write a verbal expression as an algebraic expression, we will start by identifying the inequality symbol we need to use and placing it in the center of our statement. In this case, we have the key phrase is no more than, which is represented by the symbol ≤.

On the left-hand side of the inequality, we have four times of a number m minus 2, which can be expressed as 4m-2.

On the right-hand side of the inequality, we have the quotient of a number n and 5. We can write this verbal expression as n5. Finally, we can form our inequality.


By following the same procedure as in Part A, we will first determine the inequality symbol. The key phrase for the inequality is is at least, which can be represented by ≥.

This time we have the quotient of 4 and a number s on the left-hand side.

On the right-hand side, we have the difference of a number d and 13. We can represent this expression as d-13. From here we can finally complete our inequality.


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Hint & Answer
S
Solution

LeBron James is a 2.06-meter tall American professional basketball player who, as of 2022, has played for the Cleveland Cavaliers, Miami Heat, and Los Angeles Lakers.

Lebron James
It is known that Stephen Curry, who is also an American professional basketball player, is shorter than LeBron James. Write an inequality that represents Stephen Curry's height.

To begin, we will determine which of the inequality symbols corresponds to the statement. Note that the phrase shorter than represents the inequality symbol, which means less than. Therefore, the inequality symbol is <.

Let h represent the height of Stephen Curry. From here, we can complete the left-hand side of our inequality.

Knowing that LeBron James is 2.06 meters tall, we can finally write our inequality.

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Hint & Answer
S
Solution

On a fishing trip, Zosia catches two fishes. She knows that the first fish weighs 1.3 pounds.

Fish
The second fish weighs at least 0.8 pounds more than the first fish. Write an inequality that represents the possible weights of the second fish.

To begin, we will determine which of the inequality symbols corresponds with the statement. Note that the phrase at least represents the inequality symbol, which can be expressed as ≥.

If we let w represent the weight of the second fish, we can complete the left-hand side of our inequality.

We know that the weight w is at least 0.8 pounds more than the weight of the first fish, and the first fish weighs 1.3 pounds. With this information we can now represent the verbal expression on the right-hand side algebraically.

By performing the operation on the right-hand side of the inequality, we can complete our inequality. w≥ 0.8+1.3 ⇔ w ≥ 2.1

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Hint & Answer
S
Solution

Using Inductive Reasoning to Create Inequalities

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