Section2.6Modeling with Equations and Inequalities
¶One purpose of learning math is to be able to model real-life situations and then use the model to ask and answer questions about the situation. In this lesson, we will cover the basics of modeling.
Subsection2.6.1Setting Up Equations for Models
To set up an equation modeling a real world scenario, the first thing we need to do is identify what variable we will use. The variable we use will be determined by whatever is unknown in our problem statement. Once we've identified and defined our variable, we'll use the numerical information provided to set up our equation.
Example2.6.2
A savings account starts with \(\$500\text{.}\) Each month, an automatic deposit of \(\$150\) is made. Write an equation that represents the number of months it will take for the balance to reach \(\$1{,}700\text{.}\)
To determine this equation, we'll start by making a table in order to identify a general pattern for the total amount in the account after \(m\) months:
| Months Since Saving Started |
Total Amount Saved (in Dollars) |
| \(0\) | \(500\) |
| \(1\) | \(500+150=650\) |
| \(2\) | \(500+150(2)=800\) |
| \(3\) | \(500+150(3)=950\) |
| \(4\) | \(500+150(4)=1100\) |
| \(\vdots\) | \(\vdots\) |
| \(m\) | \(500+150m\) |
Using this pattern, we can determine that an equation showing the unknown number of months, \(m\text{,}\) when the total savings equals \(\$1700\) would look like this:
\begin{equation*}
500+150m=1700
\end{equation*}
Remark2.6.4
To determine the solution to the equation in Example 2.6.2, we can continue the pattern in Table 2.6.3:
We can see that the value of \(m\) that makes the equation true is \(8\) as \(500+150(8)=1700\text{.}\) Thus it would take \(8\) months for an account starting with \(\$500\) to reach \(\$1{,}700\) if \(\$150\) is saved each month.
| Months Since Saving Started |
Total Amount Saved (in Dollars) |
| \(5\) | \(500+150(5)=1250\) |
| \(6\) | \(500+150(6)=1400\) |
| \(7\) | \(500+150(7)=1550\) |
| \(8\) | \(500+150(8)=1700\) |
In the previous example, we were able to determine the solution by creating a table and using inputs that were integers. Often the solution will not be something we are able to arrive at this way. We will need to solve for it using algebra, as we'll see in later sections. For this section, we'll only focus on setting up the equation.
Example2.6.6
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 representing when the amount of water in the bathtub will reach 6.25 ft3.
Solution
Table2.6.7Amount of Water in the Bathtub
Since this problem refers to when the amount of water will reach a certain amount, we immediately know that the unknown quantity is time. As the volume of water in the tub is measured in ft3 per minute, we know that time needs to be measured in minutes. We'll define \(t\) to be the number of minutes that water is poured into the tub. To determine this equation, we'll start by making a table of values:
| Minutes Water Has Been Poured |
Total Amount of Water (in ft3) |
| \(0\) | \(2.5\) |
| \(1\) | \(2.5+1.75=4.25\) |
| \(2\) | \(2.5+1.75(2)=6\) |
| \(3\) | \(2.5+1.75(3)=7.75\) |
| \(\vdots\) | \(\vdots\) |
| \(t\) | \(2.5+1.75t\) |
Using this pattern, we can determine that the equation representing when the amount will be 6.25 ft3 is:
\begin{equation*}
2.5+1.75t=6.25
\end{equation*}
Example2.6.8
Jakobi's annual salary as a nurse in Portland, Oregon, is \(\$73{,}290\text{.}\) His salary increased by \(4\%\) from last year. Write a linear equation modeling this scenario, where the unknown value is Jakobi's salary last year.
Solution
We need to know Jakobi's salary last year. So we'll introduce \(s\text{,}\) defined to be Jakobi's salary last year (in dollars). To determine how to set up this equation, we need to think about how he arrived at this year's salary. To get to this year's salary, his employer took last year's salary and added \(4\%\) to it. Conceptually, this means we have:
\begin{equation*}
(\text{last year's salary})+(4\%\text{ of last year's salary}) = (\text{this year's salary})
\end{equation*}
We'll represent \(4\%\) of last year's salary with \(0.04s\) since \(0.04\) is the decimal representation of \(4\%\text{.}\) This means that the equation we set up is:
\begin{equation*}
s+0.04s=73290
\end{equation*}
Example2.6.9
The price of a refrigerator after a \(15\%\) discount is \(\$612\text{.}\) Write a linear equation modeling this scenario, where the original price of the refrigerator (before the discount was applied) is the unknown quantity.
Solution
We'll let \(c\) be the original price of the refrigerator. To obtain the discounted price, we take the original price and subtract \(15\%\) of that amount. Conceptually, this looks like:
\begin{equation*}
(\text{original price})-(15\%\text{ of the original price})= (\text{discounted price})
\end{equation*}
Since the amount of the discount is \(15\%\) of the original price, we'll represent this with \(0.15c\text{.}\) The equation we set up is then:
\begin{equation*}
c-0.15c=612
\end{equation*}
Example2.6.10
A cone-shaped paper cup needs to have a volume of 6.3 in3. The radius of the cup is 1.3125 in. Write a linear equation modeling this scenario. (Hint: The volume for a cone is \(\frac{1}{3}\pi r^2 h\text{.}\))
Solution
We know that we need to find the height of this cone. So we'll introduce \(h\text{,}\) defined to be the height of the cone in inches. We know that the units need to be inches because the radius is also given in inches and the volume is given in cubic inches. Using the provided values \(r=1.3125\) and \(V=6.3\) and the formula for a cone's volume, we set this equation up as:
\begin{equation*}
\frac{1}{3}\pi (1.3125)^2 h = 6.3
\end{equation*}
Subsection2.6.2Setting Up Inequalities for Models
In general, we'll model using inequalities when we want to determine a maximum or minimum value. To identify that an inequality is needed instead of an equality, we'll look for phrases like at least, at most, at a minimum or at a maximum.
Example2.6.11
The car share company car2go has a one-time registration fee of \(\$5\) and charges \(\$14.99\) per hour for use of their vehicles. Rafael wants to use car2go and has a maximum budget of \(\$300\text{.}\) Write a linear inequality representing this scenario, where the unknown quantity is the number of hours he uses their vehicles.
Solution
We'll let \(h\) be the number of hours that Rafael uses car2go. We need the initial cost and the cost from the hourly charge to be less than or equal to \(\$300\text{,}\) which we set up as:
\begin{equation*}
5+14.99h \le 300
\end{equation*}
Example2.6.12
When an oil tank is decommissioned, it is drained of its remaining oil and then re-filled with an inert material, such as sand. A cylindrical oil tank has a volume of 275 gal and is being filled with sand at a rate of 700 gal per hour. Write a linear inequality representing this scenario, where the time it takes for the tank to overflow with sand is the unknown quantity.
Solution
The unknown in this scenario is time, so we'll define \(t\) to be the number of hours that sand is poured into the tank. (Note that we chose hours based on the rate at which the sand is being poured.) We'll represent the amount of sand poured in as \(700t\) as each hour an additional 700 gal are added. Given that we want to know when this amount exceeds 275 gal, we set this equation up as:
\begin{equation*}
700t >275
\end{equation*}
Subsection2.6.3Translating Phrases into Mathematical Expressions and Equations/Inequalities
The following table shows how to translate common phrases into mathematical expressions:
| English Phrases | Math Expressions |
| the sum of \(2\) and a number | \(x+2\) or \(2+x\) |
| \(2\) more than a number | \(x+2\) or \(2+x\) |
| a number increased by \(2\) | \(x+2\) or \(2+x\) |
| a number and \(2\) together | \(x+2\) or \(2+x\) |
| the difference between a number and \(2\) | \(x-2\) |
| the difference of \(2\) and a number | \(2-x\) |
| \(2\) less than a number | \(x-2\) (not \(2-x\)) |
| a number decreased by \(2\) | \(x-2\) |
| \(2\) decreased by a number | \(2-x\) |
| \(2\) subtracted from a number | \(x-2\) |
| a number subtracted from \(2\) | \(2-x\) |
| the product of \(2\) and a number | \(2x\) |
| twice a number | \(2x\) |
| a number times 2 | \(x\cdot 2\) or \(2x\) |
| two thirds of a number | \(\frac{2}{3}x\) |
| \(25\%\) of a number | \(0.25x\) |
| the quotient of a number and \(2\) | \(\sfrac{x}{2}\) |
| the quotient of \(2\) and a number | \(\sfrac{2}{x}\) |
| the ratio of a number and \(2\) | \(\sfrac{x}{2}\) |
| the ratio of \(2\) and a number | \(\sfrac{2}{x}\) |
We can extend this to setting up equations and inequalities. Let's look at some examples. The key is to break a complicated phrase or sentence into smaller parts, identifying key vocabulary such as “is,” “of,” “greater than,” “at most,” etc.
| English Sentences | Math Equations and Inequalities |
| The sum of \(2\) and a number is \(6\text{.}\) | \(x+2=6\) |
| \(2\) less than a number is at least \(6\text{.}\) | \(x-2\ge6\) |
| Twice a number is at most \(6\text{.}\) | \(2x\le6\) |
| \(6\) is the quotient of a number and \(2\text{.}\) | \(6=\frac{x}{2}\) |
| \(4\) less than twice a number is greater than \(10\text{.}\) | \(2x-4\gt10\) |
| Twice the difference between \(4\) and a number is \(10\text{.}\) | \(2(4-x)=10\) |
| The product of \(2\) and the sum of \(3\) and a number is less than \(10\text{.}\) | \(2(x+3)\lt10\) |
| The product of \(2\) and a number, subtracted from \(5\text{,}\) yields \(8\text{.}\) | \(5-2x=8\) |
| Two thirds of a number subtracted from \(10\) is \(2\text{.}\) | \(10-\frac{2}{3}x=2\) |
| \(25\%\) of the sum of 7 and a number is \(2\text{.}\) | \(0.25(x+7)=2\) |
SubsectionExercises
Modeling with Linear Equations and Inequalities
1
Thanh’s annual salary as a radiography technician is \({\$49{,}882.00}\text{.}\) His salary increased by \(1.8\%\) from last year. What was his salary last year?
Assume his salary last year was \(s\) dollars. Write an equation to model this scenario. There is no need to solve it.
2
A bicycle for sale costs \({\$241.73}\text{,}\) which includes \(5.1\%\) sales tax. What was the cost before sales tax?
Assume the bicycle’s price before sales tax is \(p\) dollars. Write an equation to model this scenario. There is no need to solve it.
3
The price of a washing machine after \(25\%\) discount is \({\$195.00}\text{.}\) What was the original price of the washing machine (before the discount was applied)?
Assume the washing machine’s price before the discount is \(p\) dollars. Write an equation to model this scenario. There is no need to solve it.
4
The price of a restaurant bill, including an \(13\%\) gratuity charge, was \({\$113.00}\text{.}\) What was the price of the bill before gratuity was added?
Assume the bill without gratuity is \(b\) dollars. Write an equation to model this scenario. There is no need to solve it.
5
In May 2016, the median rent price for a one-bedroom apartment in a city was reported to be \({\$801.60}\) per month. This was reported to be an increase of \(0.2\%\) over the previous month. Based on this reporting, what was the median price of a one-bedroom apartment in April 2016?
Assume the median price of a one-bedroom apartment in April 2016 was \(p\) dollars. Write an equation to model this scenario. There is no need to solve it.
6
Jon is driving an average of \(35\) miles per hour, and he is \(105\) miles away from home. After how many hours will he reach his home?
Assume Jon will reach home after \(h\) hours. Write an equation to model this scenario. There is no need to solve it.
7
Uhaul charges an initial fee of \({\$25.65}\) and then \({\$0.62}\) per mile to rent a \(15\)-foot truck for a day. If the total bill is \({\$103.77}\text{,}\) how many miles were driven?
Assume \(m\) miles were driven. Write an equation to model this scenario. There is no need to solve it.
8
Ibuprofen for infants comes in a liquid form and contains \(40\) milligrams of ibuprofen for each \(1\) milliliters of liquid. If a child is to receive a dose of \(60\) milligrams of ibuprofen, how many milliliters of liquid should they be given?
Assume \(l\) milliliters of liquid should be given. Write an equation to model this scenario. There is no need to solve it.
9
The property taxes on a \(2000\)-square-foot house are \({\$2{,}900.00}\) per year. Assuming these taxes are proportional, what are the property taxes on a \(1600\)-square-foot house?
Assume property taxes on a \(1600\)-square-foot house is \(t\) dollars. Write an equation to model this scenario. There is no need to solve it.
10
A cat litter box has a rectangular base that is \(18\) inches by \(12\) inches. What will the height of the cat litter be if \(3\) cubic feet of cat litter is poured? (Hint: \(1 \text{ ft}^3 = 1728 \text{ in}^3\))
Assume \(h\) inches will be the height of the cat litter if \(3\) cubic feet of cat litter is poured. Write an equation to model this scenario. There is no need to solve it.
11
A truck that hauls water is capable of carrying a maximum of \(2900\) lb. Water weighs \(8.3454 \frac{\text{lb}}{\text{gal}}\text{,}\) and the plastic tank on the truck that holds water weighs \(86\) lb. What’s the maximum number of gallons of water the truck can carry?
Assume the truck can carry a maximum of \(g\) gallons of water. Write an equation to model this scenario. There is no need to solve it.
12
Martha’s maximum lung capacity is \(7.4\) liters. If her lungs are full and she exhales at a rate of \(0.8\) liters per second, when will she have \(2.76\) liters of air left in his lungs?
Assume \(s\) seconds later, there would be \(2.76\) liters of air left in Martha’s lungs. Write an equation to model this scenario. There is no need to solve it.
13
A swimming pool is being filled with water from a garden hose at a rate of \(6\) gallons per minute. If the pool already contains \(100\) gallons of water and can hold \(160\) gallons, after how long will the pool overflow?
Assume \(m\) minutes later, the pool would overflow. Write an equation to model this scenario. There is no need to solve it.
14
An engineer is designing a cylindrical springform pan. The pan needs to be able to hold a volume of \(54\) cubic inches and have a diameter of \(5\) inches. What’s the minimum height it can have? (Hint: The formula for the volume of a cylinder is \(V=\pi r^2h\)).
Assume the pan’s minimum height is \(h\) inches. Write an equation to model this scenario. There is no need to solve it.
Translating English Phrases into Math Expressions and Equations
15
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(r\) to represent the unknown number.
six less than a number
16
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(t\) to represent the unknown number.
two more than a number
17
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(b\) to represent the unknown number.
the difference between a number and nine
18
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(A\) to represent the unknown number.
the sum of a number and five
19
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(B\) to represent the unknown number.
the difference between two and a number
20
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(m\) to represent the unknown number.
the difference between eight and a number
21
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(n\) to represent the unknown number.
five added to a number
22
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(q\) to represent the unknown number.
one subtracted from a number
23
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(x\) to represent the unknown number.
eight increased by a number
24
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(r\) to represent the unknown number.
five decreased by a number
25
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(t\) to represent the unknown number.
a number increased by one
26
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(b\) to represent the unknown number.
a number increased by eight
27
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(c\) to represent the unknown number.
five times a number, decreased by nine
28
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(B\) to represent the unknown number.
two times a number, increased by three
29
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(m\) to represent the unknown number.
seven less than eight times a number
30
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(n\) to represent the unknown number.
four less than twice a number
31
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(q\) to represent the unknown number.
ten less than the quotient of six and a number
32
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(x\) to represent the unknown number.
seven more than the quotient of ten and a number
33
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(r\) to represent the unknown number.
Four times a number is twelve.
34
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(t\) to represent the unknown number.
Ten times a number is seventy.
35
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(b\) to represent the unknown number.
The difference between twenty-five and a number is twenty-one.
36
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(c\) to represent the unknown number.
The sum of twenty-two and a number is thirty-four.
37
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(B\) to represent the unknown number.
The quotient of a number and thirty-nine is one thirty-ninth.
38
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(m\) to represent the unknown number.
The quotient of a number and twenty-eight is nine twenty-eighths.
39
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(n\) to represent the unknown number.
The quotient of ten and a number is five thirds.
40
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(q\) to represent the unknown number.
The quotient of twenty-six and a number is thirteen sevenths.
41
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(x\) to represent the unknown number.
The sum of five times a number and twenty-one is 236.
42
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(r\) to represent the unknown number.
The sum of three times a number and four is ninety-one.
43
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(t\) to represent the unknown number.
Two less than seven times a number is 194.
44
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(b\) to represent the unknown number.
Two less than five times a number is 238.
45
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(c\) to represent the unknown number.
The product of two and a number, increased by four, is eighty-two.
46
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(B\) to represent the unknown number.
The product of seven and a number, added to seven, is 189.
47
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(C\) to represent the unknown number.
The product of four and a number increased by three, is sixty-four.
48
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(n\) to represent the unknown number.
The product of two and a number added to six, is 104.
49
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(q\) to represent the unknown number.
one eighth of a number
50
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(x\) to represent the unknown number.
one fifth of a number
51
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(r\) to represent the unknown number.
two twenty-eighths of a number
52
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(t\) to represent the unknown number.
twenty-one thirtieths of a number
53
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(b\) to represent the unknown number.
a number decreased by three twenty-fourths of itself
54
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(c\) to represent the unknown number.
a number increased by nine thirteenths of itself
55
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(B\) to represent the unknown number.
A number decreased by two sevenths is three fifths of that number.
56
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(C\) to represent the unknown number.
A number decreased by one ninth is three halves of that number.
57
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(n\) to represent the unknown number.
Seven more than the product of three sevenths and a number yields one fifth of that number.
58
Translate the following phrase or sentence into a math expression or equation (whichever is appropriate). Use \(q\) to represent the unknown number.
Seven less than the product of two ninths and a number gives one third of that number.