Pd Modeling with Equations and Inequalities

Section 1.1 Modeling with Equations and Inequalities

Math allows you to model real situations and then use that model to answer questions about the real world. In this section, we practice setting up an equation (or inequality) from a contextualized scenario, which is a form of modeling.

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Subsection 1.1.1 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 each second passes, we gain of about $2.6$ people. The rate here is $2.6\,\frac{\text{people}}{\text{s}}\text{.}$ Note that the unit on the rate is fractional—this is common with rates.

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Example 1.1.1.

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.

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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.

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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 $150\,\frac{\text{dollar}}{\text{month}}\text{.}$

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If there’s a rate that you don’t quite understand, 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.

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Months Since Saving Started	Total Amount Saved (in Dollars)
$0$	$500$
$1$	$650$
$2$	$800$
$3$	$950$

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 from row to row. 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 $150\,\frac{\text{dollar}}{\text{month}}\text{.}$

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We spent time trying to truly understand the meaning of the numbers in this story. So now we will 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 first clearly identify what variable we will use. Since our task is to find “how much time…” and we are measuring time in months, we choose to let $m$ be our variable. That is, $m$ will stand for the number of months it will take to reach $\$1700\text{.}$ (Another good choice would be to use $t\text{,}$ since it stands for an amount of time.)

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What will the equation look like? For this example, let’s return to the table. Was there a pattern connecting the left column to the right column? 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 way, 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 $m\text{.}$

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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 1.1.2. Amount in Savings Account

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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.

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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{1.1.1} \end{equation}

and the variable represented how much the other quantity (time in this case, not the account balance) has changed. The $\pm$ should be an addition sign if the quantity is growing, and a subtraction sign if the quantity is reducing.

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Example 1.1.3.

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.

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Explanation.

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 (1.1.1).

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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.)

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Checkpoint 1.1.4.

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.

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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.

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Now using the pattern for rate modeling from (1.1.1), we have:

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\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*}

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We are not asked to solve this equation, just to write it down.

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Checkpoint 1.1.5.

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 Bulgaria’s population to reach $5$ million? Set up an equation where the solution is the number of years it will take.

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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.

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Now using the pattern for rate modeling from (1.1.1), we have:

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\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*}

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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.

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Subsection 1.1.2 Percent Applications

Here we look at some scenarios where there is an equation to set up based on percentages. An important habit to develop when doing math with percentages, is that 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.

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Example 1.1.6.

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 goal is to find Jakobi’s salary from the previous year.

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Explanation.

As soon as you are clear that the solution will be “Jakobi’s salary from last year”, that is when you should define a variable. Since it will represent a salary, we choose to use $S$ as the variable. Let $S$ represent Jakobi’s salary from last year.

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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 trying to solve this equation, just practice setting it up.

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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{1.1.2} \end{equation}

In Jakobi’s situation, his salary rose, so the $\pm$ symbol would be an addition sign. 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 subtraction sign.

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Note the variable in Jakobi’s situation is an initial value and it appears twice in the equation. This is common with percent application modeling.

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Checkpoint 1.1.7.

Kirima offered to pay the bill and tip at a restaurant where she and her friends had dinner. In total she paid $\$150$ even, where 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.

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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.

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If we let $x$ represent the original bill, we can apply (1.1.2) and we have:

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\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*}

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Again, we are not currently trying to solve this equation, just write it down.

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Example 1.1.8.

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.

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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.

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Checkpoint 1.1.9.

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.

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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:

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\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*}

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This equation’s solution will be the original price of the shirt, but for now we do not need to solve the equation.

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Subsection 1.1.3 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.

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Example 1.1.10.

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 how many hours total this semester that she could use a Zipcar vehicle.

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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.

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This exercise is much like the rate exercises from Subsection 1. 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 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 will solve it later.

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Checkpoint 1.1.11.

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.

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Explanation.

We decide to let $x$ represent the number of students enrolled at PCC in spring 2010. This is an “initial value” in the sense of (1.1.2). 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:

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\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*}

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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:

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\begin{equation*} x - 0.37x \gt 25957 \end{equation*}

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Once again, we will not try to solve this at this time.

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Reading Questions 1.1.4 Reading Questions

1.

In a word problem where there is some kind of rate, it can help you to understand the pattern if you make a .

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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?

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Exercises 1.1.5 Exercises

Review and Warmup

1.

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

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(a)

Let be the area of a house, measured in .

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(b)

Let be the age of a dog, measured in .

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(c)

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

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2.

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

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(a)

Let be the age of a person, measured in .

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(b)

Let be the distance traveled by a driver that left Portland, Oregon, bound for Boise, Idaho, measured in .

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(c)

Let be the surface area of the walls of a room, measured in .

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Applications

Modeling with Linear Equations.

Write an equation to model the scenario. There is no need to solve the equation.

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3.

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 ${380\ {\rm gal}}$ and is being filled with sand at a rate of ${530\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut hr}}}\text{.}$ 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.

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4.

A backyard swimming pool contains ${17000\ {\rm gal}}$ of water when it’s full. If it is filled at a rate of ${567\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut hr}}}\text{,}$ 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.

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5.

Ashanti is driving with an average speed of ${60\ {\rm mph}}$ on Interstate-95. She looks out the window and sees mile marker 141 pass by. How long will it take her to reach her destination, which is at mile marker 353? Write an equation to model this scenario. There is no need to solve it.

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6.

Dane filled the gas tank in his car to ${14\ {\rm gal}}\text{.}$ When the tank reaches ${1\ {\rm gal}}\text{,}$ the low gas light will come on. On average, Dane’s car uses ${0.043\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut mi}}}$ per mile driven. How many miles will Dane’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.

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7.

You planted a young tree in front of your house, and it was $6$ feet tall. Ever since, it has been growing by ${\left({\frac{1}{3}}\right)\ {\rm ft}}$ 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.

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8.

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 ${23\ {\rm psi}}\text{.}$ You have a portable device that can pump air into the tire increasing the pressure at a rate of ${1.9\ {\textstyle\frac{\rm\mathstrut psi}{\rm\mathstrut min}}}\text{.}$ How long will it take to fill the tire to the manual’s recommended pressure of ${30\ {\rm psi}}\text{?}$ Write an equation to model this scenario. There is no need to solve it.

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9.

To save up for a new phone that costs ${\$660}\text{,}$ Juliet sets aside 1% of her pay each week. This works out to ${\$4.97}$ per week. At present, she has ${\$360}$ saved up. How long will it be until Juliet has saved up enough? Write an equation to model this scenario. There is no need to solve it.

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10.

A small town would like to replace its aging water treatment system. This will cost ${\$9{,}500{,}000}\text{,}$ but the town just needs ${\$1{,}900{,}000}$ up front for downpayment on a loan that will cover the rest. The town treasury has ${\$950{,}000}$ in it already for this need, and the town can gather ${\$155{,}000}$ 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.

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11.

Neil puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at $59\,^{\circ}F$ and climbs steadily up to the boiling point of $212\,^{\circ}F\text{,}$ raising at a rate of $24\,\frac{^{\circ}F}{\text{min}}\text{.}$ How long will it take for the pot to boil? Write an equation to model this scenario. There is no need to solve it.

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12.

Sammy puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at $13\,^{\circ}C$ and climbs steadily up to the boiling point of $100\,^{\circ}C\text{,}$ raising at a rate of $15\,\frac{^{\circ}C}{\text{min}}\text{.}$ How long will it take for the pot to boil? Write an equation to model this scenario. There is no need to solve it.

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13.

Zane baked a pie at $425\,^{\circ}F$ and just took it out of the oven. It immediately starts to cool at a rate of $29\,\frac{^{\circ}F}{\text{min}}\text{.}$ How long will it take to cool to $225\,^{\circ}F\text{?}$ Write an equation to model this scenario. There is no need to solve it.

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14.

On a cold snowy day, the temperature in your home is a cozy $70\,^{\circ}F\text{,}$ but then you lose power and heating. Your home temperature begins to drop at a rate of $10\,\frac{^{\circ}F}{\text{hour}}\text{.}$ How long will it take before your home is $40\,^{\circ}F\text{?}$ Write an equation to model this scenario. There is no need to solve it.

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15.

A restaurant stocks its pantry with ${100\ {\rm lb}}$ of onions. When the supply reaches ${20\ {\rm lb}}\text{,}$ it’s time to order more. Recently, the restaurant has been using ${8\ {\rm lb}}$ 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.

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16.

At a recent trip to the casino, Esteban brought ${\$960}$ in cash. He knows he needs to hold on to ${\$90}$ in reserve to pay for dinner later. Unfortunately Esteban had rough luck and was losing money at the slot machines at an average rate of ${\$225}$ per hour. How long was Esteban gambling before he had to stop? Write an equation to model this scenario. There is no need to solve it.

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17.

For Haven’s 8th birthday party, her parents rented a venue that charges a flat fee of ${\$100}$ plus ${\$18}$ per guest. Ultimately it cost Haven’s parents ${\$334}\text{.}$ How many guests were there? Write an equation to model this scenario. There is no need to solve it.

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18.

Julien’s annual property tax had been ${\$4{,}115}\text{.}$ But he did some renovation to his house that added more square footage, and now the annual property tax is ${\$5{,}000}\text{.}$ The county assesses property tax at a rate of ${\$1.85}$ per square foot. How much area did Julien add to his home? Write an equation to model this scenario. There is no need to solve it.

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19.

Mackenzie’s current annual salary as a tax accountant is ${\$69{,}298}\text{.}$ This is with a raise of ${4.1\%}$ 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.

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20.

Nayeli’s current annual salary as an office manager is ${\$59{,}968}\text{.}$ This is with a raise of ${3\%}$ 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.

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21.

A tablet is for sale in a state where sales tax applies. The sales tax rate is ${5.8\%}$ and the total was ${\$314}\text{.}$ What was the price before sales tax? Set up an equation to answer this question. There is no need to solve it.

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22.

An annual wine subscription is for sale in a state where sales tax applies. The sales tax rate is ${4\%}$ and the total was ${\$445}\text{.}$ What was the price before sales tax? Set up an equation to answer this question. There is no need to solve it.

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23.

A dishwasher is on sale with a ${14\%}$ discount and the final price is ${\$375}\text{.}$ 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.

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24.

A cell phone is on sale with a ${22\%}$ discount and the final price is ${\$305}\text{.}$ 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.

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25.

The final bill at a restaurant one night was ${\$79.30}\text{,}$ including a ${19\%}$ 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.

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26.

The final bill at a restaurant one night was ${\$85.95}\text{,}$ including a ${17\%}$ 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.

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27.

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 ${2.9\%}$ 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.

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28.

One year, the median rent for a one-bedroom apartment in a city was reported to be ${\$1{,}490}\text{.}$ This was reported to be an increase of ${2.2\%}$ 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.

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Modeling with Linear Inequalities.

Write an inequality to model the scenario. There is no need to solve the inequality.

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29.

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 ${85\ {\rm lb}}\text{.}$ How many gallons of water can the truck haul? Set up an inequality to answer this question. There is no need to solve it.

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30.

A swimming pool is filling with water from a garden hose at a rate of ${12\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut min}}}\text{.}$ If the pool already has ${120\ {\rm gal}}$ of water and can hold up to ${390\ {\rm gal}}\text{,}$ set up an inequality modeling how much time can pass without the pool overflowing. There is no need to solve it.

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31.

Yair’s maximum lung capacity is ${7.2\ {\rm L}}\text{.}$ If his lungs are full and he exhales at a rate of ${0.8\ {\textstyle\frac{\rm\mathstrut L}{\rm\mathstrut s}}}\text{,}$ write an inequality where the solution set is the possible times when he still has at least ${0.6\ {\rm L}}$ of air left in his lungs. There is no need to solve it.

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32.

An airplane was cruising at ${34000\ {\rm ft}}\text{,}$ and then began its descent. Because of some air traffic congestion, it will descend to ${5400\ {\rm ft}}$ and then circle the airport for a while. It descends at a rate of ${620\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut min}}}\text{.}$ 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.

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33.

Dakota’s current annual salary as a paralegal is ${\$66{,}683}\text{.}$ This is with a raise by more than ${4.6\%}$ 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.

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34.

A television is on sale where all items have at least a ${5\%}$ discount. The final price is ${\$410}\text{.}$ 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.

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35.

The final bill at a restaurant one night was ${\$85.75}\text{,}$ including a tip that was at least ${17\%}\text{.}$ 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.

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36.

Overall, the median rent for a one-bedroom apartment in a city rose by ${2.7\%}$ over the previous year. Jovan was fortunate in that his rent did not raise by that high of a percentage. It went from last year’s amount to ${\$1{,}620}\text{.}$ 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.

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Challenge

37.

Last year, Joan received a ${2.5\%}$ raise. This year, she received a ${2\%}$ raise. Her current wage is ${\$10}$ an hour. What was her wage before the two raises?

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Months Since Saving Started	Total Amount Saved (in Dollars)
\(0\)	\(500\)
\(1\)	\(650\)
\(2\)	\(800\)
\(3\)	\(950\)

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