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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 $<,$ $>,$ $\leq,$ or $\geq .$ There are several ways each inequality can be phrased.

Inequality Symbol	Key Phrases
$<$	$∙ is less than or equal toter ∙ is fewer than$
$>$	$∙ is greater than or equal to ∙ is more than$
$\leq$	$∙ is less than or equal to ter ∙ is at most ∙ is no more than$
$\geq$	$∙ is greater than or equal to ∙ is at least ∙ is no less than$

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 $1 .$
$\leq$	$x + 1 \leq - 3$	The algebraic expression $x + 1$ is less than or equal to $- 3 .$
$>$	$2 x - 5 > 5$	The expression $2 x - 5$ is greater than $5 .$
$\geq$	$x \geq 2 x + 1$	The variable $x$ is greater than or equal to the expression $2 x + 1 .$

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 : \leq \geq

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

(\circ)

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.

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 .

\underline{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.

\underline{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.

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.

\underline{Red Maple} 0.31 t \underline{Cottonwood} 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.

\underline{Red Maple} 3.4 + 0.31 t \underline{Cottonwood} 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.

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 .$

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.

\underline{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.

\underline{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

$ 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

\geq

should be used.

500 + 200 m \geq 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.

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.

\underline{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.

\underline{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

\leq

symbol should be used.

129.99 + 10 t \leq 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 $y \neq < x .$

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

If $x > y,$ then $y \neq > x .$

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 — $>,$ $\leq,$ and $\geq .$

If $x > y$ and $y > z,$ then $x > z .$
If $x \leq y$ and $y \leq z,$ then $x \leq z .$
If $x \geq y$ and $y \geq z,$ then $x \geq z .$

Since this property is an axiom, it does not need proof to be accepted as true.

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Catch-Up and Review

Hint

Solution

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Solution

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Solution

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Solution