Section 1.8 Modeling with Equations and Inequalities
One reason why math is important is that it allows you to model real-life situations and then use that model to ask and answer questions about the real world. In this section, we practice setting up an equation (or inequality) from a contextualized scenario, and this is a form of modeling.
Subsection 1.8.1 Translating Phrases into Algebraic Expressions and Equations/Inequalities
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 Phrase | Math Expression |
| 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 \(2\) | \(x-2\) |
| \(2\) less than a number | \(x-2\) (not \(2-x\)) |
| a number decreased by \(2\) | \(x-2\) |
| \(2\) subtracted from a number | \(x-2\) |
| \(2\) decreased by a number | \(2-x\) |
| 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 | \(\sfrac{2}{x}\) |
Complete sentences may be communicating an equation or inequality. We suggest breaking up sentences into smaller parts. There is usually a word or phrase that corresponds to the equals sign or one of the inequality symbols. For example âisâ may correspond to an equal sign. These words/phrases are emphasized in this table.
| English Sentence | Math Equation or Inequality |
| 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 \(2\text{.}\) | \(0.25(x+7)=2\) |
Practice with translating these English phrases and sentences into math expressions, equations, and inequalities will be helpful for word problems that do have context. In the exercises for this section, you will find such practice exercises.
Subsection 1.8.2 Rate Models
A rate is a measurement that tells us how much one quantity is changing with respect to how some other quantity is changing. For example, the number of people on earth is growing by about \(2.6\) people per second. Time is passing second by second, and as that change in time happens, we have a net gain of about \(2.6\) people. The rate we are discussing is \(2.6\,\frac{\text{people}}{\text{s}}\text{.}\) Note that the unit on this rate is fractionalâthis is common with rates.
One common class of modeling applications involve rates like these. Letâs examine a first example.
Example 1.8.4.
Your savings account starts with \(\$500\text{.}\) Then each month, there is an automatic deposit of \(\$150\text{.}\) You need \(\$1700\) to afford a deposit on a new apartment. Write an equation where the solution represents how much time this will take.
Do you have an understanding of each of the numbers in this setting, and what they truly mean in context? The \(500\) is a number that only matters once in the story of this bank account: it is how much money was there when we started making automatic deposits. The \(1700\) also only matters once: at the end, we will have that much money.
But the \(150\) is different. Month after month, the account balance goes up by \(\$150\text{.}\) This number is used repeatedly in the scenario. It might look like itâs just a dollar amount, but itâs actually a rate: itâs \(150\,\frac{\text{dollar}}{\text{month}}\text{.}\)
If you have any uncertainty about understanding a rate value, it can help to make a table. We know the account balance is changing month by month, so it makes sense to track the months and the account balance.
| Months Since Saving Started |
Total Amount Saved (in Dollars) |
|---|---|
| \(0\) | \(500\) |
| \(1\) | \(650\) |
| \(2\) | \(800\) |
| \(3\) | \(950\) |
In the first column, we go up by one month from one row to the next. In the second column, we go up by \(\$150\) from one row to the next. If you are able to build a table like this, then you are dealing with a rate. In the second column, values increase by \(150\) dollars. In the first column, values increase by \(1\) month. The rate we are working with comes from dividing these: \(\frac{150\,\text{dollars}}{1\,\text{month}}\text{,}\) which is just \(150\,\frac{\text{dollar}}{\text{month}}\text{.}\)
Now that weâve spent time making sure we understand the meaning of the numbers in this story, we can try to do what we were asked to do: write an equation where the solution represents how much time it will take to reach \(\$1700\text{.}\) Itâs kind of a big deal to clearly identify what variable we will use. Our task is to write an equation where the solution represents âhow much timeâŠâ and we are measuring time in months. So one perfectly fine choice we can make is to let \(m\) be our variable, where \(m\) stands for the number of months it will take to reach \(\$1700\text{.}\) (Another common choice would be to use \(t\text{,}\) since it stands for an amount of time.)
What will the equation look like? For this example, letâs return to the table we made earlier. Was there a pattern connecting the left column to the right column? For example in the left column there is a row where \(3\) months have passed. At that time, we have \(\$950\text{.}\) How did that \(\$950\) come about? Well, we started with \(\$500\) and then added \(\$150\) not once, not twice, but three times. As an equation:
\begin{equation*}
500+150(3)=950
\end{equation*}
Thinking in this manner, we can grow the table to show this pattern and extend it to when \(m\) months have passed, even without a value in mind for \(m\text{.}\)
| Months Since Saving Started |
Total Amount Saved (in Dollars) |
|
|---|---|---|
| \(0\) | \(500\) | |
| \(1\) | \(500+150=\) | \(650\) |
| \(2\) | \(500+150(2)=\) | \(800\) |
| \(3\) | \(500+150(3)=\) | \(950\) |
| \(4\) | \(500+150(4)=\) | \(1100\) |
| \(\vdots\) | \(\vdots\) | |
| \(m\) | \(500+150m\) | |
We want there to be \(\$1700\) at the end, so apparently we want:
\begin{equation*}
500 + 150m = 1700
\end{equation*}
And that was our objective: to write down that equation. Right now we are not interested in actually solving this equation. The skill of setting up that equation is challenging enough, and this section only focuses on that setup.
There are some things worth noticing in this last example. There was a starting value for the account balance (\(\$500\)), a rate at which the account balance was changing (\(150\,\frac{\text{dollars}}{\text{month}}\)), and a final value ($1700). In the end, we had the equation
\begin{equation}
(\text{initial value}) \pm (\text{rate})\cdot\text{variable} = (\text{final value})\tag{1.8.1}
\end{equation}
and the variable represented how much the other quantity (time in this case, not the account balance) has changed. The \(\pm\) is a \(+\) if the quantity is growing, and is a \(-\) if the quantity is reducing. This is a common setup for modeling with rates.
Example 1.8.6.
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 where the solution represents when the amount of water in the bathtub will reach 6.25 ft3.
Explanation.
In this example, we have an initial amount of water (2.5 ft3), a rate at which the amount of water is changing (1.75 ft3), and a final amount of water we are going to reach (6.25 ft3). So we can use the pattern for rate modeling from (1.8.1).
We should clearly identify the variable first though. The solution is supposed to represent an amount of time. So a reasonable variable to use is \(t\text{.}\) Let \(t\) be the amount of time, in minutes, that it takes for the tub to reach 6.25 ft3. And we have the equation:
\begin{equation*}
2.5 + 1.75t = 6.25
\end{equation*}
(We are not concerned with solving this equation at this time.)
Checkpoint 1.8.7.
In the year 2020, the world population was \(7.821\) billion. And it was growing at a rate of about \(0.075\,\frac{\text{billion}}{\text{year}}\text{.}\) At this rate, how many years from 2020 will it take for the world population to reach \(9\) billion? Set up an equation where the solution is the number of years it will take.
Explanation.
Since the solution will represent an amount of time, we choose to use \(t\) as the variable. Let \(t\) be the number of years from 2020 until the world population reaches 9 billion.
Now using the pattern for rate modeling from (1.8.1), we have:
\begin{equation*}
\begin{aligned}
(\text{initial value}) + (\text{rate})\cdot\text{variable} \amp= (\text{final value})\\
7.821 + 0.075t \amp= 9
\end{aligned}
\end{equation*}
We are not asked to solve this equation, just to write it down.
Checkpoint 1.8.8.
In the year 2020, the population of Bulgaria was \(6.934\) million. And it was declining at a rate of about \(0.046\,\frac{\text{million}}{\text{year}}\text{.}\) At this rate, how many years from 2020 will it take for Bulgariaâs population to reach \(5\) million? Set up an equation where the solution is the number of years it will take.
Explanation.
Since the solution will represent an amount of time, we choose to use \(t\) as the variable. Let \(t\) be the number of years from 2020 until Bulgariaâs population reaches 5 million.
Now using the pattern for rate modeling from (1.8.1), we have:
\begin{equation*}
\begin{aligned}
(\text{initial value}) - (\text{rate})\cdot\text{variable} \amp= (\text{final value})\\
6.934 - 0.046t \amp= 5
\end{aligned}
\end{equation*}
Notice that because the population is reducing over time, we used subtraction. In effect, the rate of âgrowthâ is negative. We are not asked to solve this equation, just to write it down.
Subsection 1.8.3 Percent Applications
Section A.4 reviews the fundamentals of working with percentages. Here we look at some scenarios where there is an equation to set up based on percentages. One important consideration is that when doing math with percentages, itâs almost always best to rewrite each percentage as a decimal. For example, \(18\%\) should be written as \(0.18\) if you are going to use it to do algebra or arithmetic.
Example 1.8.9.
Jakobiâs annual salary as a nurse this year is \(\$73{,}290\text{.}\) Thatâs following a \(4\%\) raise over last yearâs salary. Write a linear equation modeling this scenario, where the solution represents Jakobiâs salary from the previous year.
Explanation.
As soon as you understand that the solution will be Jakobiâs salary from last year, that is when you should clearly define a variable. Since it will represent a salary, we choose to use \(S\) as the variable. Let \(S\) represent Jakobiâs salary from last year.
To set up the equation, we need to think about how he arrived at this yearâs salary. His employer took last yearâs salary and added \(4\%\) to that. In a diagram, we have:
\begin{equation*}
(\text{last year's salary})+(4\%\text{ of last year's salary}) = (\text{this year's salary})
\end{equation*}
We represent â\(4\%\) of last yearâs salaryâ with \(0.04S\) since \(0.04\) is the decimal equivalent to \(4\%\text{.}\) So out equation is:
\begin{equation*}
S + 0.04S = 73290
\end{equation*}
For now we are not trying to solve this equation, just practice setting it up.
We can see a pattern here that may help with future percent applications.
\begin{equation}
(\text{initial value}) \pm (\text{percent as decimal})\cdot(\text{initial value}) = (\text{final value})\tag{1.8.2}
\end{equation}
In Jakobiâs situation, his salary rose, so the \(\pm\) symbols would be a \(+\text{.}\) We can imagine other applications (like a discount at a clothing store) where the initial value is reduced, and then the \(\pm\) symbol would be a \(-\text{.}\)
The variable in Jakobiâs situation is the initial value of something. So as you can see, it appears twice in the equation. This is common with percent application modeling.
Checkpoint 1.8.10.
Kirima offered to pay the bill and tip at a restaurant where she and her friends had dinner. In total she paid \(\$150\) even, which meant the tip was a little more than \(19\%\text{.}\) Weâd like to know what was the bill before tip. Set up an equation for this situation.
Explanation.
A common mistake is to translate a question like this into âwhat is \(19\%\) of \(\$150\text{?}\)â as a way to calculate the tip amount, and then subtract that from \(\$150\text{.}\) But that is not how tipping works. The \(19\%\) is applied to the original bill, not the final total. So please read carefully.
If we let \(x\) represent the original bill, we can apply (1.8.2) and we have:
\begin{equation*}
\begin{aligned}
\pinover{x}{bill}\pinover{+}{plus}\pinover{0.19}{19\%}\pinover{\cdot}{of}\pinover{x}{bill}\amp\pinover{=}{is}\pinover{150}{\$150}
\end{aligned}
\end{equation*}
Again, we are not currently trying to solve this equation, just write it down.
Example 1.8.11.
The cost 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 solution.
Explanation.
Weâll let \(p\) be the original price of the refrigerator. To obtain the discounted price, we would have to take the original price and subtract \(15\%\) of that amount. In a diagram, we have:
\begin{equation*}
(\text{original price})-(15\%\text{ of the original price}) = (\text{discounted price})
\end{equation*}
So given the details of this refrigerator sale and our choice to use \(p\) to represent the original price, we have:
\begin{equation*}
p - 0.15p = 612
\end{equation*}
We are happy to write this equation down and do not need to solve it for now.
Checkpoint 1.8.12.
A shirt is on sale for \(20\%\) off. The sale price is \(\$51.00\text{.}\) Write an equation based on this scenario where the solution represents the shirtâs original price.
Explanation.
Let \(x\) represent the original price of the shirt. Since \(20\%\) is removed to bring the cost down to \(\$51\text{,}\) we can set up the equation:
\begin{equation*}
\begin{aligned}
\pinover{x}{original}\pinover{-}{minus}\pinover{0.20}{20\%}\pinover{\cdot}{of}\pinover{x}{original}\amp\pinover{=}{is}\pinover{51}{\$51}
\end{aligned}
\end{equation*}
This equationâs solution will be the original price of the shirt, but for now we do not need to solve the equation.
Subsection 1.8.4 Inequalities in Modeling
Occasionally an equation is not the most appropriate tool for a model, and an inequality is better. To identify when an inequality is more appropriate, look for words/phrases like âat leastâ, âat mostâ, âminimumâ, âmaximumâ, âno more thanâ, âno less thanâ, and more.
Example 1.8.13.
The car share company Zipcar has a one-time registration fee of \(\$35\) and charges \(\$9.50\) per hour for use of their vehicles. Hana wants to use Zipcar this semester and has a maximum budget of \(\$300\text{.}\) Write a linear inequality representing this scenario, where the solution set represents all the possibilities for how many hours total this semester that she could use a Zipcar vehicle.
Explanation.
The solution will represent a number of hours which is an amount of time. So we choose to use \(t\) as the variable. Let \(t\) be the number of hours that Hana drives a Zipcar this semester.
This exercise is much like the rate exercises from Subsection 2. We have an initial expense: the \(\$35\) one-time registration fee. We have a rate: \(9.50\,\frac{\text{dollar}}{\text{hour}}\text{.}\) And we have a final cost to work with: \(\$300\text{.}\) However we are not exactly trying to make Hana blow through her entire budget, so itâs not appropriate to write the equation \(35 + 9.50t = 300\text{.}\) The \(\$300\) is her maximum. Itâs totally fine (and encouraged!) for her to spend less than that. So we need to set this up using an inequality symbol. We need
\begin{align*}
(\text{registration fee}) + (\text{cost of hours driven}) \amp\leq (\text{budget maximum})\\
35 + 9.50t \amp\leq 300
\end{align*}
For now it is enough to write down this inequality, and we can work on solving it later.
Checkpoint 1.8.14.
You read on the internet that from the year 2010 to the year 2023, community colleges experienced over \(37\%\) enrollment decline. In spring 2023 at Portland Community College, there were \(25{,}957\) students. Write an inequality where the solution set represents how many students may have been at PCC in spring 2010.
Explanation.
We decide to let \(x\) represent the number of students enrolled at PCC in spring 2010. This is an âinitial valueâ in the sense of (1.8.2). But we do not want to write an equation, because we were told that community colleges experienced over a \(37\%\) enrollment decline, not exactly a \(37\%\) decline. So we write the following:
\begin{equation*}
\begin{aligned}
(\text{initial value}) \pm (\text{percent as decimal})\cdot(\text{initial value}) \amp\mathrel{?} (\text{final value})\\
x - 0.37x \amp\mathrel{?} 25957
\end{aligned}
\end{equation*}
But what inequality symbol makes sense here? If the decline was more than \(37\%\text{,}\) then \(x\) started out larger than what it would have to be for \(x - 0.37x\) to equal \(25957\text{.}\) So we need a greater than sign:
\begin{equation*}
x - 0.37x \gt 25957
\end{equation*}
Once again, we do not need to try to solve this. We are content for now to write it down.
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 this 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. Then identify what units would be most appropriate.
(a)
Let be the area of a house, measured in .
(b)
Let be the age of a dog, measured in .
(c)
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. Then identify what units would be most appropriate.
(a)
Let be the age of a person, measured in .
(b)
Let be the distance traveled by a driver that left Portland, Oregon, bound for Boise, Idaho, measured in .
(c)
Let be the surface area of the walls of a room, measured in .
Skills Practice
Translating English into Math.
Translate the phrase or sentence into a math expression or equation (whichever is appropriate).
3.
eight more than a number
4.
ten less than a number
5.
the sum of one and a number
6.
the difference between a number and two
7.
three subtracted from a number
8.
four increased by a number
9.
five decreased by a number
10.
ten times a number, decreased by four
11.
eleven times a number, increased by fourteen
12.
thirteen times a number decreased by eight
13.
fourteen times a number increased by four
14.
two more than the product of thirteen and a number
15.
three less than the product of eight and a number
16.
five more than the quotient of a number and three
17.
six less than the quotient of a number and twelve
18.
seven ninths of a number
19.
nine fourths of a number
20.
a number increased by eleven fourths of itself
21.
a number decreased by twelve thirteenths of itself
22.
fourteen less than two times a number
23.
two more than eleven times a number
24.
Fifteen more than a number is twenty-four.
25.
Twenty less than a number is fifty.
26.
Six times a number is eleven.
27.
A number divided by eight is five.
28.
The sum of thirty-three and a number is forty-eight.
29.
The difference between a number and thirty-eight is thirty-four.
30.
The product of thirteen and a number is five.
31.
The quotient of a number and fourteen is thirteen.
32.
Two more than eight times a number is ten.
33.
Three less than fourteen times a number is five.
34.
Five is the product of fourteen and six less than a number.
35.
The product of six and nine more than a number is twelve.
36.
The sum of eight and a number is the same as four times that number.
37.
Nine times a number is the same as thirteen less than twice that number.
38.
Eleven less than the quotient of a number and eight is the same as the quotient of that number and fifteen.
Applications
Modeling with Linear Equations.
Write an equation to model the scenario. There is no need to solve the equation.
39.
When a heating oil tank is decommissioned, it is drained of its remaining oil and then filled with an inert material, such as sand. One cylindrical oil tank has a volume of \({360\ {\rm gal}}\) and is being filled with sand at a rate of \({510\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut hr}}}\text{.}\) Write an equation where the solution is the amount of time, in hours, that it will take to fill the tank with sand. There is no need to solve the equation.
40.
A backyard swimming pool contains \({16300\ {\rm gal}}\) of water when itâs full. If it is filled at a rate of \({582\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut hr}}}\text{,}\) how many hours will it take to fill? Write an equation where the solution is the amount of time, in hours, that it will take to fill the pool with water. There is no need to solve the equation.
41.
Aiden is driving with an average speed of \({56\ {\rm mph}}\) on Interstate-20. They look out the window and see mile marker 100 pass by. How long will it take them to reach their destination, which is at mile marker 282? Write an equation to model this scenario. There is no need to solve it.
42.
Averie filled the gas tank in her car to \({12\ {\rm gal}}\text{.}\) When the tank reaches \({1\ {\rm gal}}\text{,}\) the low gas light will come on. On average, Averieâs car uses \({0.047\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut mi}}}\) per mile driven. How many miles will Averieâs car be able to drive before the low gas light comes on? Write an equation to model this scenario. There is no need to solve it.
43.
You planted a young tree in front of your house, and it was \(6\) feet tall. Ever since, it has been growing by \({{\frac{1}{4}}\ {\rm ft}}\) each year. How many years will it take for the tree to grow to be 10 feet tall? Write an equation to model this scenario. There is no need to solve it.
44.
One of the tires on your car looks a little flat. You measure its air pressure and are alarmed to see it so low at \({22\ {\rm psi}}\text{.}\) You have a portable device that can pump air into the tire increasing the pressure at a rate of \({1.7\ {\textstyle\frac{\rm\mathstrut psi}{\rm\mathstrut min}}}\text{.}\) How long will it take to fill the tire to the manualâs recommended pressure of \({32\ {\rm psi}}\text{?}\) Write an equation to model this scenario. There is no need to solve it.
45.
To save up for a new phone that costs \({\$780}\text{,}\) Humberto sets aside 1% of his pay each week. This works out to \({\$4.12}\) per week. At present, he has \({\$389.50}\) saved up. How long will it be until Humberto has saved up enough? Write an equation to model this scenario. There is no need to solve it.
46.
A small town would like to replace its aging water treatment system. This will cost \({\$8{,}400{,}000}\text{,}\) but the town just needs \({\$1{,}680{,}000}\) up front for downpayment on a loan that will cover the rest. The town treasury has \({\$420{,}000}\) in it already for this need, and the town can gather \({\$140{,}000}\) per month from taxes. How long will it take to reach enough for downpayment on that loan? Write an equation to model this scenario. There is no need to solve it.
47.
Malik puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at \(57\,â\) and climbs steadily up to the boiling point of \(212\,â\text{,}\) raising at a rate of \(28\,\frac{â}{\text{min}}\text{.}\) How long will it take for the pot to boil? Write an equation to model this scenario. There is no need to solve it.
48.
Pamela puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at \(17\,â\) and climbs steadily up to the boiling point of \(100\,â\text{,}\) raising at a rate of \(10\,\frac{â}{\text{min}}\text{.}\) How long will it take for the pot to boil? Write an equation to model this scenario. There is no need to solve it.
49.
Serenity baked a pie at \(425\,â\) and just took it out of the oven. It immediately starts to cool at a rate of \(27\,\frac{â}{\text{min}}\text{.}\) How long will it take to cool to \(250\,â\text{?}\) Write an equation to model this scenario. There is no need to solve it.
50.
On a cold snowy day, the temperature in your home is a cozy \(70\,â\text{,}\) but then you lose power and heating. Your home temperature begins to drop at a rate of \(10\,\frac{â}{\text{hour}}\text{.}\) How long will it take before your home is \(38\,â\text{?}\) Write an equation to model this scenario. There is no need to solve it.
51.
A restaurant stocks its pantry with \({100\ {\rm lb}}\) of onions. When the supply reaches \({30\ {\rm lb}}\text{,}\) itâs time to order more. Recently, the restaurant has been using \({7\ {\rm lb}}\) of onions per day. How long since restocking will it take until another order must be placed? Write an equation to model this scenario. There is no need to solve it.
52.
At a recent trip to the casino, Danica brought \({\$920}\) in cash. She knows she needs to hold on to \({\$100}\) in reserve to pay for dinner later. Unfortunately Danica had rough luck and was losing money at the slot machines at an average rate of \({\$240}\) per hour. How long was Danica gambling before she had to stop? Write an equation to model this scenario. There is no need to solve it.
53.
For Everettâs 8th birthday party, his parents rented a venue that charges a flat fee of \({\$90}\) plus \({\$13}\) per guest. Ultimately it cost Everettâs parents \({\$350}\text{.}\) How many guests were there? Write an equation to model this scenario. There is no need to solve it.
54.
Hugoâs annual property tax had been \({\$4{,}905}\text{.}\) But he did some renovation to his house that added more square footage, and now the annual property tax is \({\$5{,}545}\text{.}\) The county assesses property tax at a rate of \({\$2.15}\) per square foot. How much area did Hugo add to his home? Write an equation to model this scenario. There is no need to solve it.
55.
Kaedenâs current annual salary as a restaurant manager is \({\$57{,}999}\text{.}\) This is with a raise of \({2.4\%}\) over last yearâs salary. What was his salary last year? Set up an equation to answer this question. There is no need to solve it.
56.
Makenaâs current annual salary as a dental hygienist is \({\$64{,}670}\text{.}\) This is with a raise of \({5.3\%}\) over last yearâs salary. What was her salary last year? Set up an equation to answer this question. There is no need to solve it.
57.
A tool shed is for sale in a state where sales tax applies. The sales tax rate is \({6.7\%}\) and the total was \({\$472}\text{.}\) What was the price before sales tax? Set up an equation to answer this question. There is no need to solve it.
58.
A tablet is for sale in a state where sales tax applies. The sales tax rate is \({4.9\%}\) and the total was \({\$402}\text{.}\) What was the price before sales tax? Set up an equation to answer this question. There is no need to solve it.
59.
An annual wine subscription is on sale with a \({20\%}\) discount and the final price is \({\$332}\text{.}\) What was the original price before the discount is applied? Set up an equation to answer this question. There is no need to solve it.
60.
A dishwasher is on sale with a \({8\%}\) discount and the final price is \({\$463}\text{.}\) What was the original price before the discount is applied? Set up an equation to answer this question. There is no need to solve it.
61.
The final bill at a restaurant one night was \({\$73.20}\text{,}\) including a \({18\%}\) tip. What was the bill before the tip was added? Set up an equation to answer this question. There is no need to solve it.
62.
The final bill at a restaurant one night was \({\$79.85}\text{,}\) including a \({16\%}\) tip. What was the bill before the tip was added? Set up an equation to answer this question. There is no need to solve it.
63.
One year, the median rent for a one-bedroom apartment in a city was reported to be \({\$1{,}360}\text{.}\) This was reported to be an increase of \({2.5\%}\) over the previous year. Based on this reporting, what was the median rent for of a one-bedroom apartment the previous year? Set up an equation to answer this question. There is no need to solve it.
64.
One year, the median rent for a one-bedroom apartment in a city was reported to be \({\$1{,}430}\text{.}\) This was reported to be an increase of \({1.8\%}\) over the previous year. Based on this reporting, what was the median rent for of a one-bedroom apartment the previous year? Set up an equation to answer this question. There is no need to solve it.
Modeling with Linear Inequalities.
Write an inequality to model the scenario. There is no need to solve the inequality.
65.
A flatbed truck that hauls water is able to carry a maximum of \({1900\ {\rm lb}}\text{.}\) Water weighs \({8.3454\ {\textstyle\frac{\rm\mathstrut lb}{\rm\mathstrut gal}}}\text{,}\) and the plastic tank on the truck that holds the water weighs \({80\ {\rm lb}}\text{.}\) How many gallons of water can the truck haul? Set up an inequality to answer this question. There is no need to solve it.
66.
A swimming pool is filling with water from a garden hose at a rate of \({11\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut min}}}\text{.}\) If the pool already has \({130\ {\rm gal}}\) of water and can hold up to \({380\ {\rm gal}}\text{,}\) set up an inequality modeling how much time can pass without the pool overflowing. There is no need to solve it.
67.
Sammyâs maximum lung capacity is \({7.8\ {\rm L}}\text{.}\) If his lungs are full and he exhales at a rate of \({0.5\ {\textstyle\frac{\rm\mathstrut L}{\rm\mathstrut s}}}\text{,}\) write an inequality where the solution set is the possible times when he still has at least \({0.5\ {\rm L}}\) of air left in his lungs. There is no need to solve it.
68.
An airplane was cruising at \({37000\ {\rm ft}}\text{,}\) and then began its descent. Because of some air traffic congestion, it will descend to \({10000\ {\rm ft}}\) and then circle the airport for a while. It descends at a rate of \({510\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut min}}}\text{.}\) A passenger notices the plane is still descending. Set up an inequality where the solution set represents how much time might have passed since the plane started its descent. There is no need to solve it.
69.
Ashantiâs current annual salary as a web designer is \({\$71{,}385}\text{.}\) This is with a raise by more than \({2.9\%}\) over last yearâs salary. What might her salary have been last year? Set up an inequality where the solution set represents the possible answers to this question. There is no need to solve it.
70.
A cell phone is on sale where all items have at least a \({11\%}\) discount. The final price is \({\$368}\text{.}\) What might the original price have been before the discount was applied? Set up an inequality to answer this question. There is no need to solve it.
71.
The final bill at a restaurant one night was \({\$79.65}\text{,}\) including a tip that was at least \({20\%}\text{.}\) What could the bill have been before the tip was added? Set up an inequality where the solution set represents the possible amounts that the original bill might have been. There is no need to solve it.
72.
Overall, the median rent for a one-bedroom apartment in a city rose by \({1.2\%}\) over the previous year. Heath was fortunate in that his rent did not raise by that high of a percentage. It went from last yearâs amount to \({\$1{,}490}\text{.}\) Set up an inequality where the solution set represents what his rent could might have been last year. There is no need to solve it.
Challenge
73.
Last year, Joan received a \({2\%}\) raise. This year, she received a \({1.5\%}\) raise. Her current wage is \({\$11}\) an hour. What was her wage before the two raises?
