Modeling Equations and Inequalities | SAT Advanced Math | SAT Math

Introduction

The concept of *modeling *unfolded in this lesson actually includes both traditional modeling and its inverse: interpreting equations and inequalities. Math modeling usually involves understanding a real-life situation and formulating an equation that represents the relationship between the factors involved in the situation. Many SAT questions involve modeling, but some do the reverse: they provide an equation or function and ask the student to interpret some coefficient, constant, or variable in that equation or function. In both cases, an equation expresses in mathematical terms what regular prose expresses in words. We will show you examples of both modeling and interpretation in this lesson.

One further note: the generic title “equations and inequalities” is used here because you will primarily see four different types of relations on SAT modeling questions: 1) linear equations; 2) linear inequalities; 3) quadratic equations; and 4) exponential equations. This lesson on modeling is placed after the lessons in each of these four areas to ensure that you understand the basics of how these relations look and how they are simplified and solved. If you understand their mechanics well from the lessons devoted to each one, you are well-equipped to explore each of these relations further as they model real-life scenarios.

Approach Question

A certain neighborhood pool is drained every October so it can be covered for the winter. The staff has discovered that the draining process can be modeled by the function $g (t) = 18, 000 - 150 t$ , where $g (t)$ is the number of gallons left in the pool at time $t$ , in hours, after the beginning of the draining process. How many hours does it take to fully drain the pool? (Note: this is a free-response question.)

Explanation

We have used this question to begin the modeling lesson because linear equations in two variables are the most frequent type of relation modeled on the SAT. These sorts of equations are valuable for their clarity and versatility in expressing relationships between two quantities. In linear equations, the $x$ is referred to as the independent variable because it often represents something like time, which grows in quantity without depending on another factor. Conversely, the $y$ variable (or $f (x)$ or some other function notation) is called the dependent variable, since its increase or decrease depends on the passing of time or on some other regularly changing factor.

An especially helpful form of the linear equation is slope-intercept form, which you have already encountered in a previous lesson. The SAT often presents linear equations in slope intercept form because the dependent variable (y) is isolated. We can summarize and label slope-intercept form as follows:

$y = m x + b$
Variable or constant	What it represents in the equation	What it represents in the real-life scenario
y	the dependent variable	the result or outcome of the situation based on a specific input such as time
m	the slope	the rate of change
x	the independent variable	the input that determined the outcome or result
b	the y-intercept	the starting point before any change takes place

Once we know what each part of a linear equation in slope-intercept form represents, we can readily translate any information given in such an equation. Our function in this case is $g (t) = 18, 000 - 150 t$ . That doesn’t look exactly like slope-intercept form, so it’s helpful to consider how to rearrange the function. Since we know by the commutative property of addition that any two terms added together can be reordered without affecting the outcome, and since the $- 150 t$ here can also be understood as $+ (- 150 t)$ , let’s rewrite the function as $g (t) = - 150 t + 18, 000$ . Here is what each part tells us:

$g (t)$ : the number of gallons left in the pool (information provided in the question)

$- 150$ : the rate of change over time; this means the pool is losing 150 gallons every hour

$t$ : the time, in hours, since the draining began (information provided in the question)

$18, 000$ : the starting point; this means the pool must have held 18,000 gallons of water before the draining began

We have technically done more than is necessary to answer the question, but we want you to be equipped for questions that ask about any of the four major aspects of a linear equation. In this case, as we use the UnCLES method well and focus on what the question is asking for, we can concentrate on the variable t, since the question “how many hours …?” is asking about time. But how do we assign a value for $g (t)$ so we can solve for $t$ ? As you identified key terms in the question, you may have noticed the phrase “fully drained.” If you understand that the draining affects the number of gallons remaining, you can then infer that “fully drained” must mean zero gallons remaining. We therefore plug in $0$ for $g (t)$ , and the equation becomes $0 = 18, 000 - 150 t$ . You’ve had plenty of practice solving equations as you’ve practiced for the SAT with Achievable, so we’ll trust you to walk through the steps. The answer is 12.

Definitions

Independent Variable: A variable that is manipulated or controlled in an experiment or scenario to understand its effects on another variable (known as the dependent variable). The overwhelmingly most common independent variable on the SAT is time. The independent variable is represented by $x$ in a function.
Dependent Variable: A variable that is observed in an experiment or scenario to understand how it changes based on the variation in a controlled or dependable variable (known as the independent variable). The dependent variable is represented by $y$ in a function.

Topics for Cross-Reference

Variations

The closest thing to a “variation” in this lesson is inequalities, because (unlike all the other relations modeled on the SAT) they are not equations. As such, they have their own language: “greater than”, “less than”, “maximum”, “minimum”. Keep an eye out for such terms and make sure you are comfortable with the difference between an inequality and an equation.

Strategy Insights

One of the reasons the slope-intercept form of a linear equation is so handy is that it helps us distinguish between the variable term ( $m x$ ) and the constant term ( $b$ ). In a real-life situation, this can make all the difference; $x$ is changing over time and is multiplied by the rate of change, while $b$ doesn’t change (and probably represents the starting point of the action). The question will often ask about one or the other of these two terms, so the distinction between variable and constant terms is vital.

Flashcard Fodder

Take a break from adding flashcards based on this lesson …. but consider this a great opportunity to review the ones you already have!

Sample Questions

Difficulty 1

A student sets a homework plan whereby she will work $45$ minutes per day plus $25$ minutes per unscheduled hour after school from 4:00 p.m. - 10: p.m. Which of the following equations models the student’s plan, where $x$ is the number of unscheduled hours in the given range and $y$ is the total number of minutes allotted to homework?

A. $y = 45 + 25 x$
B. $y = 25 + 45 x$
C. $y = 70 x$
D. $y = 70$

(spoiler)

The answer is $y = 45 + 25 x$ . The table presented in the explanation of the Approach question serves us well here. Consider also the Strategy Insight: it’s vital to distinguish between the variable term and the constant term in a linear equation in slope-intercept form. We can answer this question by identifying which factor in the real-life scenario doesn’t change and which one does. The student is going to invest $45$ minutes in homework no matter what, so $45$ must be the constant. Meanwhile, the $25$ minutes contributes to the total time only when we know how many hours, $x$ , she has free in the six-hour range given. So $25 x$ must be the other term. Even though the right answer inverts the mx term and the $b$ term from typical slope-intercept form, it’s the only one containing $25 x$ and $45$ as it should. Notice the two answers containing $70$ ; these are traps set for those who mistakenly think that $25$ and $45$ can be added together in this scenario. Since $45$ and $25 x$ are not like terms, they cannot be added together.

Difficulty 2

A boy pushes a water balloon off of his bedroom windowsill with hopes of drenching his unsuspecting brother, who is sitting on the ground below. The flight of the water balloon is modeled by the function $h (t) = - 4.9 t^{2} + 16$ , where $h (t)$ is the height of the balloon, in feet, above the ground and $t$ is the time, in seconds, since the balloon left the windowsill. If this function were graphed, which of the following would be the best interpretation of the $y$ -intercept?

A. The initial height of the balloon above the ground was 4.9 feet.
B. The height of the balloon 4.9 seconds after beginning flight was 16 feet above the ground.
C. The height of the balloon as it sat on the windowsill was 16 feet above the ground.
D. The height of the balloon 16 seconds after beginning flight was 4.9 feet above the ground.

(spoiler)

The answer is The height of the balloon as it sat on the windowsill was 16 feet above the ground.. Thus far in this lesson, we have been dealing with linear equations, but the lesson on quadratics should be fresh in your memory. As with a linear equation in slope-intercept form, a quadratic equation in standard form reveals the $y$ -intercept directly in its constant term. If you understand that, you can identify the $y$ -intercept as $16$ right away.

All that remains is to figure out the significance of $16$ in this real-life situation. Just as with linear equations, the $y$ -intercept of a quadratic function points to the “starting point” if we limit our domain to nonnegative values. Since the independent variable here is time, a negative value wouldn’t make sense anyway; meanwhile, we know that the $y$ -intercept occurs at time zero. This is the point at which the balloon is pushed off the sill; at that point, it is still $16$ feet above the ground.

If you have any doubt, remember that you can test your answer by plugging in $0$ for $x$ , since the question pertains to the $y$ -intercept, where $x$ must equal $0$ . If we plug in $0$ for $x$ , the first term zeroes out and we have simply $h (t) = 16$ . That means the height must be $16$ at the $y$ -intercept.

Difficulty 3

A middle school design committee is drawing up plans for a school with sufficient classroom space to serve 375 students. The design includes large classrooms that can accommodate 25 students and small classrooms that can accommodate 15 students. If the equation $25 x + 15 y = 375$ is used to model this situation, which of the following is the best interpretation of $y$ in this context?

A. The total number of students accommodated by the large classrooms
B. The total number of students accommodated by the small classrooms
C. The number of large classrooms
D. The number of small classrooms

(spoiler)

The answer is The number of small classrooms. This linear equation, unlike those previously seen in this lesson, is in standard form rather than slope-intercept form. The virtue of standard form is that it immediately reveals the quantities multiplied by each variable. In the coefficients $25$ and $15$ , we understand that $x$ must represent something multiplied by $25$ and $y$ must represent something multiplied by $15$ . If we read the situation carefully, we can infer that $x$ must therefore represent the number of large classrooms and $y$ the number of small classrooms.

One interesting way to test this answer would be to figure out how many classrooms there would be if the committee decided to go with only small classrooms. In that case, we would divide $375$ by $15$ , arriving at $25$ small classrooms. Plugging in $25$ for $y$ in the equation, while substituting $0$ for $x$ (because in this scenario there are no large classrooms), we can see that the equation works.

Difficulty 4

The maximum value of $a$ is $9$ less than $4$ times another number $b$ . Which of the following inequalities identifies the range of possible values of $a$ ?

A. $a \leq 9 - 4 b$
B. $a < 9 - 4 b$
C. $a \leq 4 b - 9$
D. $a \geq 4 b - 9$

(spoiler)

The answer is $a \leq 4 b - 9$ . Our first inequality of the lesson! The answer choices, of course, give away that we are dealing with an inequality, but, in using the UnCLES method to identify key terms, did you notice the word that suggests an inequality? That word is “maximum”. And although the answer choices reveal that we’re dealing with an inequality, they do not specify whether the phrase “or equal to” should be included in our thinking. It rests upon us to interpret the word “maximum” and realize that the word does not exclude the idea of being equal to the maximum value. So we must choose an answer with the line under the inequality indicating “less than or equal to.” But we must also find an answer using “less than or equal to” (not “greater than or equal to”), because “maximum” implies all values equal to or less than the given value. Two answers are already eliminated.

Continuing to use the UnCLES method and keeping the remaining answer choices in mind, we need only choose between $9 - 4 b$ and $4 b - 9$ . Both choices correctly interpret “ $4$ times another number $b$ ” as $4 b$ . But how do we understand “ $9$ less than”? Perhaps surprisingly, the $9$ goes after the $4 b$ ; this makes sense because “less than” means it is subtracted from the $4 b$ . So $4 b - 9$ is the correct expression.

Difficulty 5

A scientist observing yeast growth in a lab proposes a function to model the growth: $Y (t) = Y_{0} (3^{k t})$ , where $y (t)$ is the total number of yeast cells, $Y_{0}$ is the initial number of yeast cells, $t$ is the time in hours, and $k$ is a growth constant. If the experiment begins with 50,000 yeast cells and there are 150,000 yeast cells after 4 hours, what is the value of k, according to the scientist’s model?

A. 1/50,000
B. 1/4
C. 4
D. 50,000

(spoiler)

The answer is $\frac{1}{4}$ . There are a number of factors involved in this situation; while it is helpful to understand the factors, in the end we can plug all the given values into the given function and find ourselves with only one unknown remaining. So the quickest approach here is to identify which values plug in for which unknowns in the function. A summary:

$y (t)$ =final amount= $150, 000$

$Y_{0}$ =initial amount= $50, 000$

$k$ =growth constant=unknown

$t$ =time= $4$ hours

After substituting all values, our equation is $150, 000 = 50, 000 (3^{4 k})$ . Dividing both sides by $50, 000$ yields $3 = 3^{4 k}$ . How to proceed from here?

Remember that any number or variable is implicitly raised to the first power, because anything raised to that power is unchanged. $1^{1} = 1$ , $2^{1} = 2$ , etc. So we can rewrite the left side of the equation as $3^{1}$ . This helps apply the principle of equal bases in an equation. That is, if $3^{1} = 3^{4 k}$ , we can simply drop the bases because they are identical. It must be the case that the exponents on each side of the equation are equal to each other, so $1 = 4 k$ . Hence our answer of $\frac{1}{4}$ .

For Reflection

How will you approach modeling questions on test day? Write down at least three takeaways from this module.
Rate the difficulty of these questions for you from 1 (no problem) to 5 (problem!). This will help you decide when to answer them and when to skip them on test day.
Do you understand the difference between modeling a real-life situation with an equation and interpreting an equation by translating it into words? There are both kinds of problems in this lesson; consider reviewing whichever of the two kinds you feel less confident about.

Introduction

Approach Question

A certain neighborhood pool is drained every October so it can be covered for the winter. The staff has discovered that the draining process can be modeled by the function $g (t) = 18, 000 - 150 t$ , where $g (t)$ is the number of gallons left in the pool at time $t$ , in hours, after the beginning of the draining process. How many hours does it take to fully drain the pool? (Note: this is a free-response question.)

Explanation

$y = m x + b$
Variable or constant	What it represents in the equation	What it represents in the real-life scenario
y	the dependent variable	the result or outcome of the situation based on a specific input such as time
m	the slope	the rate of change
x	the independent variable	the input that determined the outcome or result
b	the y-intercept	the starting point before any change takes place

$g (t)$ : the number of gallons left in the pool (information provided in the question)

$- 150$ : the rate of change over time; this means the pool is losing 150 gallons every hour

$t$ : the time, in hours, since the draining began (information provided in the question)

$18, 000$ : the starting point; this means the pool must have held 18,000 gallons of water before the draining began

Definitions

Independent Variable: A variable that is manipulated or controlled in an experiment or scenario to understand its effects on another variable (known as the dependent variable). The overwhelmingly most common independent variable on the SAT is time. The independent variable is represented by $x$ in a function.
Dependent Variable: A variable that is observed in an experiment or scenario to understand how it changes based on the variation in a controlled or dependable variable (known as the independent variable). The dependent variable is represented by $y$ in a function.

Topics for Cross-Reference

Variations

Strategy Insights

Flashcard Fodder

Take a break from adding flashcards based on this lesson …. but consider this a great opportunity to review the ones you already have!

Sample Questions

Difficulty 1

A student sets a homework plan whereby she will work $45$ minutes per day plus $25$ minutes per unscheduled hour after school from 4:00 p.m. - 10: p.m. Which of the following equations models the student’s plan, where $x$ is the number of unscheduled hours in the given range and $y$ is the total number of minutes allotted to homework?

A. $y = 45 + 25 x$
B. $y = 25 + 45 x$
C. $y = 70 x$
D. $y = 70$

(spoiler)

Difficulty 2

A boy pushes a water balloon off of his bedroom windowsill with hopes of drenching his unsuspecting brother, who is sitting on the ground below. The flight of the water balloon is modeled by the function $h (t) = - 4.9 t^{2} + 16$ , where $h (t)$ is the height of the balloon, in feet, above the ground and $t$ is the time, in seconds, since the balloon left the windowsill. If this function were graphed, which of the following would be the best interpretation of the $y$ -intercept?

A. The initial height of the balloon above the ground was 4.9 feet.
B. The height of the balloon 4.9 seconds after beginning flight was 16 feet above the ground.
C. The height of the balloon as it sat on the windowsill was 16 feet above the ground.
D. The height of the balloon 16 seconds after beginning flight was 4.9 feet above the ground.

(spoiler)

Difficulty 3

A middle school design committee is drawing up plans for a school with sufficient classroom space to serve 375 students. The design includes large classrooms that can accommodate 25 students and small classrooms that can accommodate 15 students. If the equation $25 x + 15 y = 375$ is used to model this situation, which of the following is the best interpretation of $y$ in this context?

A. The total number of students accommodated by the large classrooms
B. The total number of students accommodated by the small classrooms
C. The number of large classrooms
D. The number of small classrooms

(spoiler)

Difficulty 4

The maximum value of $a$ is $9$ less than $4$ times another number $b$ . Which of the following inequalities identifies the range of possible values of $a$ ?

A. $a \leq 9 - 4 b$
B. $a < 9 - 4 b$
C. $a \leq 4 b - 9$
D. $a \geq 4 b - 9$

(spoiler)

Difficulty 5

A scientist observing yeast growth in a lab proposes a function to model the growth: $Y (t) = Y_{0} (3^{k t})$ , where $y (t)$ is the total number of yeast cells, $Y_{0}$ is the initial number of yeast cells, $t$ is the time in hours, and $k$ is a growth constant. If the experiment begins with 50,000 yeast cells and there are 150,000 yeast cells after 4 hours, what is the value of k, according to the scientist’s model?

A. 1/50,000
B. 1/4
C. 4
D. 50,000

(spoiler)

$y (t)$ =final amount= $150, 000$

$Y_{0}$ =initial amount= $50, 000$

$k$ =growth constant=unknown

$t$ =time= $4$ hours

After substituting all values, our equation is $150, 000 = 50, 000 (3^{4 k})$ . Dividing both sides by $50, 000$ yields $3 = 3^{4 k}$ . How to proceed from here?

For Reflection

How will you approach modeling questions on test day? Write down at least three takeaways from this module.
Rate the difficulty of these questions for you from 1 (no problem) to 5 (problem!). This will help you decide when to answer them and when to skip them on test day.
Do you understand the difference between modeling a real-life situation with an equation and interpreting an equation by translating it into words? There are both kinds of problems in this lesson; consider reviewing whichever of the two kinds you feel less confident about.