10. Using Inductive Reasoning to Create Inequalities
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Chapter 1
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 |
Using Inductive Reasoning to Create Inequalities
Slide of 9
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 leastthat 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 or equal toter∙ is fewer than
|
| > | ∙ is greater than or equal to∙ is more than
|
| ≤ | ∙ is less than or equal toter∙ 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. |
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.
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.
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.
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.
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.
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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 Air1.2t
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 Left12−1.2t
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 than5.1 liters, the inequality is strict and the symbol
>should be used.
12−1.2t>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.
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.
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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 Maple0.31tCottonwood0.64t
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 Maple3.4+0.31tCottonwood2.7+0.64t
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 thanthe that of the cottonwood. Therefore, the inequality will be strict and the
<symbol should be used.
3.4+0.31t<2.7+0.64t
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.
After some research online, Tadeo learns that a 1967 Chevrolet Impala is available in his town for about $23000. 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 $25000 in case of any price changes before he can afford it. Write an inequality that models when Tadeo will have saved at least $25000.
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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 Savings200m
Since he starts with $500, his total savings after m months can be represented by the following expression.
Total Savings500+200m
To write the inequality, the next step will be determining the inequality sign. Remember that he will need at least$25000. This means that he will need either $25000 or more than $25000. Since the possibility of an equality exists, the inequality will be non-strict and the symbol
≥should be used.
500+200m≥25000
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.
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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 Fare10t
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 Cost129.99+10t
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+10t≤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
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.
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.
Using Inductive Reasoning to Create Inequalities
Exercises
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.
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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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What inequality represents the verbal expression?
|
A number t plus 5 is less than 11. |
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|
7 is more than twice a number k minus 1. |
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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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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. |
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|
The quotient of 4 and a number s is at least the difference of a number d and 13. |
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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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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.
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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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On a fishing trip, Zosia catches two fishes. She knows that the first fish weighs 1.3 pounds.
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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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Using Inductive Reasoning to Create Inequalities
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