ORCCA Modeling with Equations and Inequalities

Section 2.8 Modeling with Equations and Inequalities

One purpose of learning math is to be able to model real-life situations and then use the model to ask and answer questions about the situation. In this lesson, we will examine the basics of modeling to set up an equation (or inequality).

Subsection 2.8.1 Setting Up Equations for Rate Models

To set up an equation modeling a real world scenario, the first thing we need to do is identify what variable we will use. The variable we use will be determined by whatever is unknown in our problem statement. Once we've identified and defined our variable, we'll use the numerical information provided to set up our equation.

Example 2.8.1.

A savings account starts with $\$500\text{.}$ Each month, an automatic deposit of $\$150$ is made. Write an equation that represents the number of months it will take for the balance to reach $\$1{,}700\text{.}$

Explanation

To determine this equation, we might start by making a table in order to identify a general pattern for the total amount in the account after $m$ months.

Using this pattern, we can determine that an equation showing the unknown number of months, $m\text{,}$ when the total savings equals $\$1700$ would look like this:

\begin{equation*} 500+150m=1700 \end{equation*}

Table 2.8.2. Amount in Savings Account

Months Since Saving Started	Total Amount Saved (in Dollars)
$0$	$500$
$1$	$500+150=650$
$2$	$500+150(2)=800$
$3$	$500+150(3)=950$
$4$	$500+150(4)=1100$
$\vdots$	$\vdots$
$m$	$500+150m$

Remark 2.8.3.

To determine the solution to the equation in Example 2.8.1, we could continue the pattern in Table 2.8.2:

We can see that the value of $m$ that makes the equation true is $8$ as $500+150(8)=1700\text{.}$ Thus, it would take $8$ months for an account starting with $\$500$ to reach $\$1{,}700$ if $\$150$ is saved each month.

Table 2.8.4. Amount in Savings Account

Months Since Saving Started	Total Amount Saved (in Dollars)
$5$	$500+150(5)=1250$
$6$	$500+150(6)=1400$
$7$	$500+150(7)=1550$
$8$	$500+150(8)=1700$

Here we are able to determine the solution by creating a table and using inputs that were whole numbers. Often the solution will not be something we can find this way. We will need to solve the equation using algebra, as we'll learn how to do in later sections. For this section, we'll only focus on setting up the equation.

Example 2.8.5.

A bathtub contains 2.5 ft³ of water. More water is being poured in at a rate of 1.75 ft³ per minute. Write an equation representing when the amount of water in the bathtub will reach 6.25 ft³.

Explanation

Since this problem refers to when the amount of water will reach a certain amount, we immediately know that the unknown quantity is time. As the volume of water in the tub is measured in ft³ per minute, we know that time needs to be measured in minutes. We'll define $t$ to be the number of minutes that water is poured into the tub. To determine this equation, we'll start by making a table of values:

Table 2.8.6. Amount of Water in the Bathtub

Minutes Water Has Been Poured	Total Amount of Water (in ft³)
$0$	$2.5$
$1$	$2.5+1.75=4.25$
$2$	$2.5+1.75(2)=6$
$3$	$2.5+1.75(3)=7.75$
$\vdots$	$\vdots$
$t$	$2.5+1.75t$

Using this pattern, we can determine that the equation representing when the amount will be 6.25 ft³ is:

\begin{equation*} 2.5+1.75t=6.25 \end{equation*}

Subsection 2.8.2 Setting Up Equations for Percent Problems

Section 2.7 reviewed some basics of working with percentages, and even solved some one-step equations that were set up using percentages. Here we look at some scenarios where there is an equation to set up based on percentages, but it is slightly more involved than a one-step equation.

Example 2.8.7.

Jakobi's annual salary as a nurse in Portland, Oregon, is $\$73{,}290\text{.}$ His salary increased by $4\%$ from last year. Write a linear equation modeling this scenario, where the unknown value is Jakobi's salary last year.

Explanation

We need to know Jakobi's salary last year. So we'll introduce $s\text{,}$ defined to be Jakobi's salary last year (in dollars). To set up the equation, we need to think about how he arrived at this year's salary. To get to this year's salary, his employer took last year's salary and added $4\%$ to it. Conceptually, this means we have:

\begin{equation*} (\text{last year's salary})+(4\%\text{ of last year's salary}) = (\text{this year's salary}) \end{equation*}

We'll represent $4\%$ of last year's salary with $0.04s$ since $0.04$ is the decimal representation of $4\%\text{.}$ This means that the equation we set up is:

\begin{equation*} s+0.04s=73290 \end{equation*}

Checkpoint 2.8.8.

Example 2.8.9.

The price of a refrigerator after a $15\%$ discount is $\$612\text{.}$ Write a linear equation modeling this scenario, where the original price of the refrigerator (before the discount was applied) is the unknown quantity.

Explanation

We'll let $c$ be the original price of the refrigerator. To obtain the discounted price, we take the original price and subtract $15\%$ of that amount. Conceptually, this looks like:

\begin{equation*} (\text{original price})-(15\%\text{ of the original price})= (\text{discounted price}) \end{equation*}

Since the amount of the discount is $15\%$ of the original price, we'll represent this with $0.15c\text{.}$ The equation we set up is then:

\begin{equation*} c-0.15c=612 \end{equation*}

Checkpoint 2.8.10.

Subsection 2.8.3 Setting Up Equations for Geometry Problems

With geometry problems and algebra, there is often the possibility to draw some picture to help understand the scenario better. Additionally it is often necessary to rely on some formula from geometry, such as the formulas from Subsection 2.2.1.

Example 2.8.11.

An Olympic-size swimming pool is rectangular and 50 m in length. We don't know its width, but we do know that it required 150 m of painter's tape to outline the edge of the pool during recent renovations. Use this information to set up an equation that models the width of the pool.

Explanation

Since the pool's shape is a rectangle, it helps to sketch a rectangle representing the pool as in Figure 2.8.12. Since we know its length is 50 m, it is a good idea to label that in the sketch. The width is our unknown quantity, so we can use $w$ as a variable to represent the pool's width in meters and label that too.

Figure 2.8.12. An Olympic-size pool

Since it required 150 m of painter's tape to outline the pool, we know the perimeter of the pool is 150 m. This suggests using the perimeter formula for a rectangle: $P=2(\ell+w)\text{.}$ (This formula was discussed in Subsection 2.2.1).

With this formula, we can substitute $150$ in for $P$ and $50$ in for $\ell\text{:}$

\begin{equation*} 150=2(50+w) \end{equation*}

and this equation models the width of the pool.

Checkpoint 2.8.13.

Subsection 2.8.4 Modeling with Linear Inequalities

In general, we'll model using inequalities when we want to determine a maximum or minimum value. To identify that an inequality is needed instead of an equality, we'll look for phrases like at least, at most, at a minimum or at a maximum.

Example 2.8.14.

The car share company car2go has a one-time registration fee of $\$5$ and charges $\$14.99$ per hour for use of their vehicles. Hana wants to use car2go and has a maximum budget of $\$300\text{.}$ Write a linear inequality representing this scenario, where the unknown quantity is the number of hours she uses their vehicles.

Explanation

We'll let $h$ be the number of hours that Hana uses car2go. We need the initial cost and the cost from the hourly charge to be less than or equal to $\$300\text{,}$ which we set up as:

\begin{equation*} 5+14.99h \le 300 \end{equation*}

Example 2.8.15.

When an oil tank is decommissioned, it is drained of its remaining oil and then re-filled with an inert material, such as sand. A cylindrical oil tank has a volume of 275 gal and is being filled with sand at a rate of 700 gal per hour. Write a linear inequality representing this scenario, where the time it takes for the tank to overflow with sand is the unknown quantity.

Explanation

The unknown in this scenario is time, so we'll define $t$ to be the number of hours that sand is poured into the tank. (Note that we chose hours based on the rate at which the sand is being poured.) We'll represent the amount of sand poured in as $700t$ as each hour an additional 700 gal are added. Given that we want to know when this amount exceeds 275 gal, we write this inequality as:

\begin{equation*} 700t >275 \end{equation*}

Note that we were only asked to write the inequality to represent the given scenario. So this inequality is our answer. If we were asked to find the time it takes the tank to overflow with sand, then we would use the skills taught in Section 2.6 to solve the inequality, which would help us determine that answer.

Subsection 2.8.5 Translating Phrases into Mathematical Expressions and Equations/Inequalities

The following table shows how to translate common phrases into mathematical expressions:

Table 2.8.16. Translating English Phrases into Math Expressions

English Phrases	Math Expressions
the sum of $2$ and a number	$x+2$ or $2+x$
$2$ more than a number	$x+2$ or $2+x$
a number increased by $2$	$x+2$ or $2+x$
a number and $2$ together	$x+2$ or $2+x$
the difference between a number and $2$	$x-2$
the difference of $2$ and a number	$2-x$
$2$ less than a number	$x-2$ (not $2-x$)
a number decreased by $2$	$x-2$
$2$ decreased by a number	$2-x$
$2$ subtracted from a number	$x-2$
a number subtracted from $2$	$2-x$
the product of $2$ and a number	$2x$
twice a number	$2x$
a number times 2	$x\cdot 2$ or $2x$
two thirds of a number	$\frac{2}{3}x$
$25\%$ of a number	$0.25x$
the quotient of a number and $2$	$\sfrac{x}{2}$
the quotient of $2$ and a number	$\sfrac{2}{x}$
the ratio of a number and $2$	$\sfrac{x}{2}$
the ratio of $2$ and a number	$\sfrac{2}{x}$

We can extend this to setting up equations and inequalities. Let's look at some examples. The key is to break a complicated phrase or sentence into smaller parts, identifying key vocabulary such as “is,” “of,” “greater than,” “at most,” etc.

Table 2.8.17. Translating English Sentences into Math Equations

English Sentences	Math Equations and Inequalities
The sum of $2$ and a number is $6\text{.}$	$x+2=6$
$2$ less than a number is at least $6\text{.}$	$x-2\ge6$
Twice a number is at most $6\text{.}$	$2x\le6$
$6$ is the quotient of a number and $2\text{.}$	$6=\frac{x}{2}$
$4$ less than twice a number is greater than $10\text{.}$	$2x-4\gt10$
Twice the difference between $4$ and a number is $10\text{.}$	$2(4-x)=10$
The product of $2$ and the sum of $3$ and a number is less than $10\text{.}$	$2(x+3)\lt10$
The product of $2$ and a number, subtracted from $5\text{,}$ yields $8\text{.}$	$5-2x=8$
Two thirds of a number subtracted from $10$ is $2\text{.}$	$10-\frac{2}{3}x=2$
$25\%$ of the sum of 7 and a number is $2\text{.}$	$0.25(x+7)=2$

Exercises 2.8.6 Exercises

Review and Warmup

1.

Identify a variable you might use to represent each quantity. And identify what units would be most appropriate.

Let be the area of a house, measured in .
Let be the age of a dog, measured in .
Let be the amount of time passed since a driver left Seattle, Washington, bound for Portland, Oregon, measured in .

2.

Identify a variable you might use to represent each quantity. And identify what units would be most appropriate.

Let be the age of a person, measured in .
Let be the distance traveled by a driver that left Portland, Oregon, bound for Boise, Idaho, measured in .
Let be the surface area of the walls of a room, measured in .

Modeling with Linear Equations

3.

Ashley’s annual salary as a radiography technician is ${\$59{,}566.00}\text{.}$ Her salary increased by $2.7\%$ from last year. What was her salary last year?

Assume her salary last year was $s$ dollars. Write an equation to model this scenario. There is no need to solve it.

4.

Randi’s annual salary as a radiography technician is ${\$31{,}620.00}\text{.}$ His salary increased by $2\%$ from last year. What was his salary last year?

Assume his salary last year was $s$ dollars. Write an equation to model this scenario. There is no need to solve it.

5.

A bicycle for sale costs ${\$84.32}\text{,}$ which includes $5.4\%$ sales tax. What was the cost before sales tax?

Assume the bicycle’s price before sales tax is $p$ dollars. Write an equation to model this scenario. There is no need to solve it.

6.

A bicycle for sale costs ${\$118.25}\text{,}$ which includes $7.5\%$ sales tax. What was the cost before sales tax?

Assume the bicycle’s price before sales tax is $p$ dollars. Write an equation to model this scenario. There is no need to solve it.

7.

The price of a washing machine after $15\%$ discount is ${\$119.00}\text{.}$ What was the original price of the washing machine (before the discount was applied)?

Assume the washing machine’s price before the discount is $p$ dollars. Write an equation to model this scenario. There is no need to solve it.

8.

The price of a washing machine after $5\%$ discount is ${\$161.50}\text{.}$ What was the original price of the washing machine (before the discount was applied)?

Assume the washing machine’s price before the discount is $p$ dollars. Write an equation to model this scenario. There is no need to solve it.

9.

The price of a restaurant bill, including an $18\%$ gratuity charge, was ${\$70.80}\text{.}$ What was the price of the bill before gratuity was added?

Assume the bill without gratuity is $b$ dollars. Write an equation to model this scenario. There is no need to solve it.

10.

The price of a restaurant bill, including an $14\%$ gratuity charge, was ${\$79.80}\text{.}$ What was the price of the bill before gratuity was added?

Assume the bill without gratuity is $b$ dollars. Write an equation to model this scenario. There is no need to solve it.

11.

In May 2016, the median rent price for a one-bedroom apartment in a city was reported to be ${\$1{,}402.80}$ per month. This was reported to be an increase of $0.2\%$ over the previous month. Based on this reporting, what was the median price of a one-bedroom apartment in April 2016?

Assume the median price of a one-bedroom apartment in April 2016 was $p$ dollars. Write an equation to model this scenario. There is no need to solve it.

12.

In May 2016, the median rent price for a one-bedroom apartment in a city was reported to be ${\$1{,}512.00}$ per month. This was reported to be an increase of $0.8\%$ over the previous month. Based on this reporting, what was the median price of a one-bedroom apartment in April 2016?

Assume the median price of a one-bedroom apartment in April 2016 was $p$ dollars. Write an equation to model this scenario. There is no need to solve it.

13.

Fabrienne is driving an average of $30$ miles per hour, and she is $63$ miles away from home. After how many hours will she reach her home?

Assume Fabrienne will reach home after $h$ hours. Write an equation to model this scenario. There is no need to solve it.

14.

Aleric is driving an average of $34$ miles per hour, and he is $34$ miles away from home. After how many hours will he reach his home?

Assume Aleric will reach home after $h$ hours. Write an equation to model this scenario. There is no need to solve it.

15.

Uhaul charges an initial fee of ${\$25.00}$ and then ${\$0.77}$ per mile to rent a $15$-foot truck for a day. If the total bill is ${\$152.05}\text{,}$ how many miles were driven?

Assume $m$ miles were driven. Write an equation to model this scenario. There is no need to solve it.

16.

Uhaul charges an initial fee of ${\$27.20}$ and then ${\$0.63}$ per mile to rent a $15$-foot truck for a day. If the total bill is ${\$75.71}\text{,}$ how many miles were driven?

Assume $m$ miles were driven. Write an equation to model this scenario. There is no need to solve it.

17.

A cat litter box has a rectangular base that is $18$ inches by $24$ inches. What will the height of the cat litter be if $4.5$ cubic feet of cat litter is poured? (Hint: $1 \text{ ft}^3 = 1728 \text{ in}^3$)

Assume $h$ inches will be the height of the cat litter if $4.5$ cubic feet of cat litter is poured. Write an equation to model this scenario. There is no need to solve it.

18.

A cat litter box has a rectangular base that is $18$ inches by $18$ inches. What will the height of the cat litter be if $2.25$ cubic feet of cat litter is poured? (Hint: $1 \text{ ft}^3 = 1728 \text{ in}^3$)

Assume $h$ inches will be the height of the cat litter if $2.25$ cubic feet of cat litter is poured. Write an equation to model this scenario. There is no need to solve it.

Modeling with Linear Inequalities

19.

A truck that hauls water is capable of carrying a maximum of $1900$ lb. Water weighs $8.3454 \frac{\text{lb}}{\text{gal}}\text{,}$ and the plastic tank on the truck that holds water weighs $85$ lb. Assume the truck can carry a maximum of $g$ gallons of water. Write an inequality to model this scenario. There is no need to solve it.

20.

A truck that hauls water is capable of carrying a maximum of $3000$ lb. Water weighs $8.3454 \frac{\text{lb}}{\text{gal}}\text{,}$ and the plastic tank on the truck that holds water weighs $91$ lb. Assume the truck can carry a maximum of $g$ gallons of water. Write an inequality to model this scenario. There is no need to solve it.

21.

Phil’s maximum lung capacity is $7.7$ liters. If his lungs are full and he exhales at a rate of $0.8$ liters per second, write an inequality that models when he will still have at least ${4.9}$ liters of air left in his lungs. There is no need to solve it.

22.

Renee’s maximum lung capacity is $4.1$ liters. If her lungs are full and she exhales at a rate of $0.8$ liters per second, write an inequality that models when she will still have at least ${1.22}$ liters of air left in her lungs. There is no need to solve it.

23.

A wading pool is being filled with water from a garden hose at a rate of $10$ gallons per minute. If the pool already contains $20$ gallons of water and can hold up to $140$ gallons, set up an inequality modeling how much time can pass without the pool overflowing. There is no need to solve it.

24.

A wading pool is being filled with water from a garden hose at a rate of $8$ gallons per minute. If the pool already contains $30$ gallons of water and can hold up to $190$ gallons, set up an inequality modeling how much time can pass without the pool overflowing. There is no need to solve it.

25.

An engineer is designing a cylindrical springform pan (the kind of pan a cheesecake is baked in). The pan needs to be able to hold a volume at least $70$ cubic inches and have a diameter of $8$ inches. Write an inequality modeling possible height of the pan. There is no need to solve it.

26.

An engineer is designing a cylindrical springform pan (the kind of pan a cheesecake is baked in). The pan needs to be able to hold a volume at least $212$ cubic inches and have a diameter of $10$ inches. Write an inequality modeling possible height of the pan. There is no need to solve it.