ORCCA Modeling with Equations and Inequalities

Section 1.8 Modeling with Equations and Inequalities

Objectives: PCC Course Content and Outcome Guide

MTH 60 CCOG 1.3
MTH 60 CCOG 2.5

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).

Figure 1.8.1. Alternative Video Lesson

Subsection 1.8.1 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 1.8.2.

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 set up an 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. In Figure 1.8.3, we find the pattern is that after $m$ months, the total amount saved is $500+150m\text{.}$

Using this pattern, we determine that an equation representing when the total savings equals $\$1700$ is:

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

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

Figure 1.8.3. Amount in Savings Account

Right now we are not interested in actually solving this equation and finding the number of months $m$ until the savings has reached $\$1700\text{.}$ The skill of setting up that equation is challenging enough, and this section only focuses on that setup.

Example 1.8.4.

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:

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

Figure 1.8.5. Amount of Water in the Bathtub

Using this pattern, we 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 1.8.2 Setting Up Equations for Percent Problems

Section A.4 reviews some basics of working with percentages, and even solves some one-step equations that are set up using percentages. Here we look at some scenarios where there is an equation to set up based on percentages, but it things are a little more complicated than with a one-step equation. One important fact is that when doing math with percentages, we always start by rewriting the percentage as a decimal. For example, $18\%$ should be written as $0.18$ if you are going to us it to do algebra or arithmetic.

Example 1.8.6.

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 want to work with 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 1.8.7.

Example 1.8.8.

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 1.8.9.

Subsection 1.8.3 Setting Up Inequalities for Models

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 1.8.10.

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 1.8.11.

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 set this equation up as:

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

Subsection 1.8.4 Translating Phrases into Algebraic Expressions and Equations/Inequalities

Void of context, there are certain short phrases and expressions in English that have mathematical meaning, and might show up in a modeling scenario. The following table shows how to translate some of these common phrases into algebraic 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	$\sfrac{2}{x}$

Table 1.8.12. Translating English Phrases into Math Expressions

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.

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 $2\text{.}$	$0.25(x+7)=2$

Table 1.8.13. Translating English Sentences into Math Equations

A certain amount of practice with translating these English phrases and sentences into math expressions, equations, and inequalities can be helpful for word problems that do have context. In the exercises for this section, you will find such practice exercises.

Reading Questions 1.8.5 Reading Questions

1.

It is common to come across a word problem where there is some kind of rate. In a problem like that, it can help you to understand the pattern if you make a .

2.

It is common to come across a word problem where some percent is either added or subtracted from an unknown original value. With the approach described in theis section for setting up an equation, how many times will you use the variable in such an equation?

3.

Is there any difference between these three phrases, or do they all mean the same thing?

ten subtracted from a number
ten less than a number
ten minus a number

Exercises 1.8.6 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.

Sherial’s annual salary as a radiography technician is ${\$38{,}494.00}\text{.}$ Her salary increased by $1.3\%$ 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.

Ken’s annual salary as a radiography technician is ${\$42{,}066.00}\text{.}$ His salary increased by $2.6\%$ 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 ${\$180.71}\text{,}$ which includes $6.3\%$ 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 ${\$210.60}\text{,}$ which includes $5.3\%$ 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 $25\%$ discount is ${\$172.50}\text{.}$ 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.

8.

The price of a washing machine after $15\%$ discount is ${\$221.00}\text{.}$ 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.

9.

The price of a restaurant bill, including an $10\%$ gratuity charge, was ${\$110.00}\text{.}$ 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 $17\%$ gratuity charge, was ${\$11.70}\text{.}$ 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 ${\$904.50}$ per month. This was reported to be an increase of $0.5\%$ 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 ${\$1{,}002.00}$ 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.

13.

Briana is driving an average of $41$ miles per hour, and she is $123$ miles away from home. After how many hours will she reach his home?

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

14.

Ryan is driving an average of $44$ miles per hour, and he is $83.6$ miles away from home. After how many hours will he reach his home?

Assume Ryan 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 ${\$31.90}$ and then ${\$0.89}$ per mile to rent a $15$-foot truck for a day. If the total bill is ${\$182.31}\text{,}$ 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 ${\$34.10}$ and then ${\$0.75}$ per mile to rent a $15$-foot truck for a day. If the total bill is ${\$94.85}\text{,}$ 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 $24$ inches by $12$ inches. What will the height of the cat litter be if $3$ 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 $3$ 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 $24$ inches by $24$ inches. What will the height of the cat litter be if $4$ 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 $4$ cubic feet of cat litter is poured. Write an equation to model this scenario. There is no need to solve it.

19.

Ibuprofen for infants comes in a liquid form and contains $30$ milligrams of ibuprofen for each $0.75$ milliliters of liquid. If a child is to receive a dose of $50$ milligrams of ibuprofen, how many milliliters of liquid should they be given?

Assume $l$ milliliters of liquid should be given. Write an equation to model this scenario. There is no need to solve it.

20.

Ibuprofen for infants comes in a liquid form and contains $35$ milligrams of ibuprofen for each $0.875$ milliliters of liquid. If a child is to receive a dose of $60$ milligrams of ibuprofen, how many milliliters of liquid should they be given?

Assume $l$ milliliters of liquid should be given. Write an equation to model this scenario. There is no need to solve it.

21.

The property taxes on a $2400$-square-foot house are ${\$2{,}832.00}$ per year. Assuming these taxes are proportional, what are the property taxes on a $1600$-square-foot house?

Assume property taxes on a $1600$-square-foot house is $t$ dollars. Write an equation to model this scenario. There is no need to solve it.

22.

The property taxes on a $1900$-square-foot house are ${\$2{,}489.00}$ per year. Assuming these taxes are proportional, what are the property taxes on a $1500$-square-foot house?

Assume property taxes on a $1500$-square-foot house is $t$ dollars. Write an equation to model this scenario. There is no need to solve it.

Modeling with Linear Inequalities

23.

A truck that hauls water is capable of carrying a maximum of $1800$ lb. Water weighs $8.3454 \frac{\text{lb}}{\text{gal}}\text{,}$ and the plastic tank on the truck that holds water weighs $74$ 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.

24.

A truck that hauls water is capable of carrying a maximum of $2800$ lb. Water weighs $8.3454 \frac{\text{lb}}{\text{gal}}\text{,}$ and the plastic tank on the truck that holds water weighs $80$ 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.

25.

Lindsay’s maximum lung capacity is $6.8$ 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 ${0.56}$ liters of air left in his lungs. There is no need to solve it.

26.

Lily’s maximum lung capacity is $7.3$ 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 ${3.94}$ liters of air left in his lungs. There is no need to solve it.

27.

A swimming pool is being filled with water from a garden hose at a rate of $9$ gallons per minute. If the pool already contains $100$ gallons of water and can hold up to $334$ gallons, set up an inequality modeling how much time can pass without the pool overflowing. There is no need to solve it.

28.

A swimming pool is being filled with water from a garden hose at a rate of $7$ gallons per minute. If the pool already contains $10$ gallons of water and can hold up to $108$ gallons, set up an inequality modeling how much time can pass without the pool overflowing. There is no need to solve it.

29.

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 $33$ cubic inches and have a diameter of $6$ inches. Write an inequality modeling possible height of the pan. There is no need to solve it.

30.

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 $96$ cubic inches and have a diameter of $7$ inches. Write an inequality modeling possible height of the pan. There is no need to solve it.