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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.
Figure 1.8.1. Alternative Video Lesson

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.
Table 1.8.2. Translating English Phrases into Math 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 “is” may corrrespond to an equals sign. These words/phrases are emphasized in this table.
Table 1.8.3. Translating English Sentences into Math Equations
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 fractional—this is common with rates.
One common class of modeling applications involve rates like these. Let’s 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 it’s just a dollar amount, but it’s actually a rate: it’s \(150\,\frac{\text{dollar}}{\text{month}}\text{.}\)
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 we’ve 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 \(\$1700\text{.}\) It’s kind of a big deal to clearly identify what variable we will use. Our task is to write an equation where the solution represents “how much time…” 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, let’s 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)
\(0\) \(500\)
\(1\) \(500+150=\) \(650\)
\(2\) \(500+150(2)=\) \(800\)
\(3\) \(500+150(3)=\) \(950\)
\(4\) \(500+150(4)=\) \(1100\)
\(\vdots\) \(\vdots\)
\(m\) \(500+150m\)
Figure 1.8.5. Amount in Savings Account
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 most 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 shrinking 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 Bulgaria’s 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 Bulgaria’s 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 “growth” is negative. We are not asked to solve this equation, just to write it down.

Subsection 1.8.3 Percent Applications

[cross-reference to target(s) "section-percentages" missing or not unique] 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, it’s 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.

Jakobi’s annual salary as a nurse this year is \(\$73{,}290\text{.}\) That’s following a \(4\%\) raise over last year’s salary. Write a linear equation modeling this scenario, where the solution represents Jakobi’s salary from the previous year.
Explanation.
As soon as you understand that the solution will be Jakobi’s 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 variablbe. Let \(S\) represent Jakobi’s salary from last year.
To set up the equation, we need to think about how he arrived at this year’s salary. His employer took last year’s 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 “\(4\%\) of last year’s salary” 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 tryign 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 Jakobi’s sutuation, 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 Jakobi’s 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 \(19\%\text{.}\) We’d 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 “what is \(19\%\) of \(\$150\text{?}\)” 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.
We’ll 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 shirt’s 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 equation’s 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 “at least”, “at most”, “minimum”, “maximum”, “no more than”, “no less than”, 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 reperesnt a number of hours which is an amount of time. So we choose to use \(t\) as the variable. Let \(t\) be the nubmer 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 exacly trying to make Hana blow through her entire budget, so it’s not appropriate to write the equation \(35 + 9.50t = 300\text{.}\) The \(\$300\) is her maximum. It’s 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 enumber of students enrolled at PCC in spring 2010. This is an “initial value” 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.
seven more than a number
4.
nine less than a number
5.
the sum of ten and a number
6.
the difference between a number and one
7.
two subtracted from a number
8.
three increased by a number
9.
four decreased by a number
10.
eight times a number, decreased by eleven
11.
ten times a number, increased by six
12.
eleven times a number decreased by fifteen
13.
thirteen times a number increased by ten
14.
fourteen more than the product of five and a number
15.
two less than the product of fifteen and a number
16.
three more than the quotient of a number and ten
17.
five less than the quotient of a number and six
18.
five twelfths of a number
19.
seven tenths of a number
20.
a number increased by nine fifths of itself
21.
a number decreased by eleven fourths of itself
22.
thirteen less than eight times a number
23.
fourteen more than four times a number
24.
Eleven more than a number is forty-three.
25.
Sixteen less than a number is twenty-nine.
26.
Five times a number is three.
27.
A number divided by six is twelve.
28.
The sum of twenty-nine and a number is twenty-seven.
29.
The difference between a number and thirty-three is thirteen.
30.
The product of eleven and a number is twelve.
31.
The quotient of a number and thirteen is six.
32.
Fourteen more than six times a number is two.
33.
Two less than twelve times a number is eleven.
34.
Three is the product of seven and four less than a number.
35.
The product of five and fifteen more than a number is nine.
36.
The sum of six and a number is the same as eleven times that number.
37.
Eight times a number is the same as five less than twice that number.
38.
Nine less than the quotient of a number and fifteen is the same as the quotient of that number and twelve.

Applications

Modeling with Linear Equations.
Write an equation to model the scenario. There is no need to solve the equation.
39.
When an 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 \({335\ {\rm gal}}\) and is being filled with sand at a rate of \({560\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut hr}}}\text{.}\) 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 \({15500\ {\rm gal}}\) of water when its full. If it is filled at a rate of \({705\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut hr}}}\text{,}\) 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.
Seth is driving with an average speed of \({64\ {\rm mph}}\) on Interstate-5. He looks out the window and sees mile marker 178 pass by. How long will it take him to reach his destination, which is at mile marker 383? Write an equation to model this scenario. There is no need to solve it.
42.
Alan filled the gas tank in his car to \({15\ {\rm gal}}\text{.}\) When the tank reaches\({1\ {\rm gal}}\text{,}\) the low gas light will come on. On average, Alan’s car uses \({0.045\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut mi}}}\) per mile driven. How many miles will Alan’s 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 \(4\) feet tall. Ever since, it has been growing by \({{\frac{1}{6}}\ {\rm ft}}\) each year. How many years will it take for the tree to grow to be 12 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 \({25\ {\rm psi}}\text{.}\) You have a portable device that can pump air into the tire increasing the pressure at a rate of \({1.4\ {\textstyle\frac{\rm\mathstrut psi}{\rm\mathstrut min}}}\text{.}\) How long will it take to fill the tire to the manual’s recommended pressure of \({31\ {\rm psi}}\text{?}\) Write an equation to model this scenario. There is no need to solve it.
45.
To save up for a new phone that costs \({\$735}\text{,}\) Ezra sets aside 1% of their pay each week. This works out to \({\$6.03}\) per week. At present, they have \({\$372.50}\) saved up. How long will it be until Ezra 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 \({\$5{,}800{,}000}\text{,}\) but the town just needs \({\$1{,}160{,}000}\) up front for downpayment on a loan that will cover the rest. The town treasury has \({\$290{,}000}\) in it already for this need, and the town can gather \({\$125{,}000}\) 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.
Kailey puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at \(61\,℉\) and climbs steadily up to the boiling point of \(212\,℉\text{,}\) raising at a rate of \(26\,\frac{℉}{\text{min}}\text{.}\) 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.
Mariana puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at \(14\,℃\) and climbs steadily up to the boiling point of \(100\,℃\text{,}\) raising at a rate of \(15\,\frac{℃}{\text{min}}\text{.}\) 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.
Paola baked a pie at \(425\,℉\) and just took it out of the oven. It immediately starts to cool at a rate of \(21\,\frac{℉}{\text{min}}\text{.}\) How long will it take to cool to \(225\,℉\text{?}\) 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 \(70\,℉\text{,}\) but then you lose power and heating. Your home temperature begins to drop at a rate of \(15\,\frac{℉}{\text{hour}}\text{.}\) How long will it take before your home is \(43\,℉\text{?}\) 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 \({25\ {\rm lb}}\text{,}\) it’s 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, Averie brought \({\$800}\) in cash. She knows she needs to hold on to \({\$100}\) in reserve to pay for dinner later. Unfortunately Averie had rough luck and was losing money at the slot machines at an average rate of \({\$235}\) per hour. How long was Averie gambling before she had to stop? Write an equation to model this scenario. There is no need to solve it.
53.
For Darion’s 8th birthday party, his parents rented a venue that charges a flat fee of \({\$110}\) plus \({\$12}\) per guest. Utimately it cost Darion’s parents \({\$314}\text{.}\) How many guests were there? Write an equation to model this scenario. There is no need to solve it.
54.
Ezekiel’s annual property tax had been \({\$4{,}375}\text{.}\) But he did some renovation to his house that added more square footage, and now the annual property tax is \({\$4{,}875}\text{.}\) The county assesses property tax at a rate of \({\$1.98}\) per square foot. How much area did Ezekiel add to his home? Write an equation to model this scenario. There is no need to solve it.
55.
Humberto’s current annual salary as a radiography technician is \({\$71{,}297}\text{.}\) This is with a raise of \({5.5\%}\) over last year’s salary. What was his salary last year? Set up an equation to answer this question. There is no need to solve it.
56.
Kailey’s current annual salary as a web designer is \({\$61{,}966}\text{.}\) This is with a raise of \({4.4\%}\) over last year’s salary. What was her salary last year? Set up an equation to answer this question. There is no need to solve it.
57.
A designer dress is for sale in a state where sales tax applies. The sales tax rate is \({6.2\%}\) and the total was \({\$367}\text{.}\) What was the price before sales tax? Set up an equation to answer this question. There is no need to solve it.
58.
A tool shed is for sale in a state where sales tax applies. The sales tax rate is \({4.4\%}\) and the total was \({\$498}\text{.}\) What was the price before sales tax? Set up an equation to answer this question. There is no need to solve it.
59.
A tablet is on sale with a \({16\%}\) discount and the final price is \({\$428}\text{.}\) 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 bicycle is on sale with a \({25\%}\) discount and the final price is \({\$358}\text{.}\) 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 \({\$66.75}\text{,}\) including a \({20\%}\) 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 \({\$73.40}\text{,}\) 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.
63.
One year, the median rent for a one-bedroom apartment in a city was reported to be \({\$1{,}300}\text{.}\) This was reported to be an increase of \({1.4\%}\) 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 \({\$1{,}370}\text{.}\) This was reported to be an increase of \({2.8\%}\) 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 \({1800\ {\rm lb}}\text{.}\) 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 \({90\ {\rm lb}}\text{.}\) 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 \({9\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut min}}}\text{.}\) If the pool already has \({120\ {\rm gal}}\) of water and can hold up to \({370\ {\rm gal}}\text{,}\) set up an inequality modeling how much time can pass without the pool overflowing. There is no need to solve it.
67.
Pamela’s maximum lung capacity is \({7.6\ {\rm L}}\text{.}\) If her lungs are full and she exhales at a rate of \({0.9\ {\textstyle\frac{\rm\mathstrut L}{\rm\mathstrut s}}}\text{,}\) write an inequality where the solution set is the possible times when she still has at least \({0.8\ {\rm L}}\) of air left in her lungs. There is no need to solve it.
68.
An airplane was cruising at \({36000\ {\rm ft}}\text{,}\) and then began its descent. Because of some air traffic congestion, it will descend to \({9500\ {\rm ft}}\) and then cirlce the airport for a while. It descends at a rate of \({760\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut min}}}\text{.}\) A passenger notices the planee is still descending. Set up an inequality where thee solution set represents how much time might have passed since the plane started its descent. There is no need to solve it.
69.
Zoey’s current annual salary as a microelectronics engineer is \({\$68{,}682}\text{.}\) This is with a raise by more than \({2\%}\) over last year’s 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 dishwasher is on sale where all items have at least a \({8\%}\) discount. The final price is \({\$463}\text{.}\) 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 \({\$73.20}\text{,}\) including a tip that was at least \({18\%}\text{.}\) 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 \({2.9\%}\) over the previous year. Evelyn was fortunate in that her rent did not raise by that high of a percentage. It went from last year’s amount to \({\$1{,}170}\text{.}\) Set up an inequality where the solution set represents what her 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 \({3\%}\) raise. Her current wage is \({\$10}\) an hour. What was her wage before the two raises?
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