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College Algebra - Final
Milestone Exam with
Answers! Latest Edition
2024
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2 questions were answered incorrectly
...
The quotient of a positive and negative
number is always negative
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The product of two negative numbers is
always positive
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This is incorrect
...
The correct quotient
is -9
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The product of two positive numbers is
always positive
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CONCEPT
Multiplying and Dividing Positive and Negative Numbers

2

Select the solution to the following system of equations:

•

•

correct
•

•

RATIONALE

Before you can add equations, sometimes you
must multiply entire equations by a scalar value for
the method to work as intended
...

Here, by multiplying

by , the result

contains a
term, which will cancel when
combined with the second equation
...

When we add the two equations, the
cancel, leaving


...


Once we divide both sides by

in for

term will

, we get a


...


Let's use the second equation


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Subtracting a negative

is the same as adding a

postive
...

equals


...
com)


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This is the solution to the system of equations
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[4 + 8 × (5 – 3)] ÷ 5 + 6
•

10 correct
•

10
...
8
RATIONALE
Following the Order of Operations, we must first
evaluate everything in parentheses and grouping
symbols
...
Here, we must evaluate
5 minus 3 first
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There are still operations inside
grouping symbols to evaluate
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Next, we add 4 and 16 to complete the
operations inside parentheses
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Now there is just division and
subtraction
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20 divided by 5 is 4
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4 plus 6 is 10
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What do we know about the graph of this quadratic equation,
based on its formula?
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The vertex is

and it opens upward
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•

•

The vertex is

and it opens upward
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•

The vertex is

and it opens downward
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The values of and in particular
give us useful information about the graph
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If is positive, the parabola opens
upward
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In this case, since is negative, we know the
parabola opens downward
...


and

to find the x-coordinate of the

The values and can be plugged into this formula
to give us the x-coordinate of the vertex
...
Simplify the denominator
...
Evaluate the
division to get the x-coordinate of the vertex
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The x-coordinate of the vertex is

...

Return to the original equation, but write in the
calculated x-coordinate, , for every instance of
...

squared is
...

is

times is
subtraction
...
Next, evaluate

times
...
Finally, evaluate the addition and

This is the y-coordinate to the parabola's vertex
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CONCEPT
Introduction to Parabolas

5

Suppose
Find the value of

and


...


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Evaluate
in

for the variable

in the function

Once has been replaced with
expression
...


, evaluate the

evaluates to
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Once has been replaced with , evaluate the
expression
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CONCEPT
Function of a Function

6

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•

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•

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Then we will identify the
intersection point
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Re-writing each equation in the form
will help us easily identify the yintercept and the slope to graph each line
...
First, subtract from both
sides
...
Then divide both sides
to isolate
...
Simplify the

right side by dividing each term by
...



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Connecting these two points makes
this line
...

The goal is to isolate the variable onto
one side of the equals sign
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Then divide both sides by
to isolate
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Simplify the

right side by dividing

by


...
This

tells us that the y-intercept is at
the slope is

and


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Connecting these two points makes
this line
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Coach Trevor asks his players to run from one corner to the
other corner by running diagonally across the field
...

•

115 meters
•

190 meters
•

136 meters correct
•

185 meters
RATIONALE

We can use the Pythagorean Theorem to calculate the
length of a diagonal
...
First,
substitute 80 for a and 10 for b
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Once we have the given values plugged into the
Pythagorean Theorem, we can evaluate the exponents
...
Now we
can add these values together
...
Finally, we can take
the square root of both sides to find the value of c
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When we have a squared term, such as , taking the
square root of both sides will cancel this operation
...
The
distance from one corner of the field to the other,
rounded to the nearest meter, is 136 meters
...
coli infections reported in a
certain neighborhood in a given week
...

If the number of infections continues to grow exponentially, what
will the number of infections be in week 6?
•

456 people correct
•

203 people

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We will use information from the problem to find
values to plug into this equation
...
The infection rate is

, so this is the value

, so this is our value for

(remember to write it as a decimal)
...
We will need to solve for
...

plus

is


...


to the power of

is


...

There will be 456 people infected in week 6
...


Find the value of the following expression
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com)

•

•

correct
•

•

RATIONALE
This question involves several properties of
logarithms
...


This means we can express

as


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Then, the Power Property of
Logs states that exponents inside a logarithm can
be expressed as outside scalar multiples of the
logarithm
...
Finally, we can substitute

the values we were previously given for
, and

,


...

Once these given values are substituted into the
expression, simplify each term and then perform
the addition and subtraction
...


times

is
...


CONCEPT
Applying Properties of Logarithms

10

The spread of a disease can be modeled as
, where
N is the number of infected people, and t is time (in days)
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How long will it take until the number of infected people reaches
1,400?
•

28 days
•

98 days
•

49 days correct
•

14 days
RATIONALE
The spread of the disease is modeled by this equation
...

Once N is replaced, we can solve for t
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On the left, 1400 divided by 200 is 7
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On the left, 7 squared is 49
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CONCEPT
Solving Multi-step Equations

11

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A
protein molecule has an overall length of 3000 times (or
times) the diameter of a hydrogen atom
...


When multiplying numbers in scientific
notation, you must deal with the numbers
and 10s separately
...
05 and
3
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1
...
15
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-10 plus 3 equals -7, which is the exponent
for the base 10
...

The length of the protein molecule (in
meters), when written in scientific notation,
is


...

•

•

•

correct
•

RATIONALE

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Multiply the first terms

, the outside terms

, the inside terms

, and the last

terms

...

The expression

multiplies to

...


and

can combine to

simplify the last term,


...
Recall that the

is

equivalent to
, which can be substituted in
our expression
...
Next, evaluate


...
Finally, combine like
and
...
com)

can be written

...

Which of the following equations is equivalent to 4y – 1 = 15, but
written only in terms of x?
•

20x – 12 = 14
•

5x – 12 = 16
•

20x – 12 = 16 correct
•

5x – 12 = 14
RATIONALE
To find an equivalent equation written in terms of x we
can use the second equation y = 5x – 3
...

Now we have an equation in terms of x
...

4 times 5x equals 20x, and 4 times -3 equals -12
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15 plus 1 equals 16
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CONCEPT
Substitution in multi-step linear equations

14

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Find the slope of this function
...

The slope is the difference in coordinates from the two
points divided by the difference in coordinates from the
same two points
...


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In the numerator, the difference in coordinates is
minus

, which is the same as

denominator, the difference in
, which is is the same as
the line is

plus , or
...
The slope of


...
The following table shows a projection of the population,

,

after years
...
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•

9
...
8 years
•

10
...
4 years
RATIONALE
in general, exponential decay is modeled using this
equation
...

The initial population is

, so this is the value

for
...
We
want to know how many years until the population
is
, which will be our value for
...
Start by simplifying what's
inside the parentheses
...
Next, divide both sides by


...
To undo the variable exponent, take
the log of both sides
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is now isolated
...

and


...


It will take 10
...


CONCEPT
Exponential Decay

16

Geri runs a total of 3
...

How many feet is this equivalent to?
•

4,989 feet
•

5,280 feet
•

15,840 feet

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A conversion factor is a fraction
with equal quantities in the numerator and
denominator, but written with different units
...
We know how many feet are
in 1 mile
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There are 5280 feet in 1 mile
...
1 miles
into feet, multiply

by the fraction


...
The units of
miles will cancel, leaving only feet
...

In the numerator, 3
...
3
...


CONCEPT
Converting Units

17

Find the solution for x if I7x + 5I = 19
...
com)

•

•

•

correct
•

RATIONALE
To solve this equation, we will need to create two
separate equations without absolute value bars
...
The second
equation must consider the case when the expression
has the opposite value
...

To solve for x, start by subtracting 5 from both sides
...
On the right, 19 minus 5 is
14
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The same
process can be applied to the second equation 7x + 5 =
-19
...

On the left, we have only 7x
...
Then divide both sides by 7 to isolate x
...


CONCEPT
Absolute Value Equations

18

Consider the following table
...
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•

correct

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•

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Remember, the
value indicates the distance the point
moves left (negative) or right (positive) from
the origin,

...


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For

, move one point to the left and

six points up from
For


...


For

, move three points to the left

and eight points up from
For


...


CONCEPT
Graph of a Line

19

The vertical position

(in feet) of a rock seconds after it was

dropped from a cliff is given by the formula
The base of the cliff corresponds to


...


After how many seconds will the rock hit the ground at the base of
the cliff?
•

5 seconds correct

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When a quadratic equation is set equal to zero,
you can use the quadratic formula to solve for
...

In this case,

,

in the

, and


...


, begin

We can start with the denominator, since it is
easily simplified by multiplying and

...

squared equals
times
between

, and

times

equals
...


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Now
that the expression underneath the radical is
simplified, take the square root of


...
To find the
solutions, create two equations (due to the )
...
Next, evaluate
the numerator in each equation
...


In the second fraction, plus

equals


...


The two solutions to the quadratic equation are
seconds and

seconds
...

It will take seconds for the rock to hit the
ground at the base of the cliff
...
com)

•

•

•

•

correct
RATIONALE
In order to cancel like terms, we have to first factor
both the numerator and denominator, and then look
for common factors in the top and bottom of the
fraction
...

To factor this quadratic, we need to identify a pair
of integers whose product is the constant term ( )
and whose sum is the coefficient of the x-term ( )
...
This means that and
integer pair for this quadratic
...
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The original numerator can be written as

...

Again, we need to identify a pair of integers whose
product is the constant term (

) and whose sum

is the coefficient of the x-term ( )
...
This means that
integer pair for this quadratic
...
Now we can rewrite the original
fraction with the new factored numerator and
denominator
...
To simplify, we cancel factors that appear in
both the top and the bottom
...
The rational

can be simplified to

CONCEPT
Simplifying Rational Expressions

21

For the arithmetic sequence beginning with the terms {-2, 0, 2, 4,
6, 8
...
com)


...

To find the value of any nth term, use the formula
, where
is the value of the first
term,
is the value of the nth term, d is the common
difference, and n is the number of terms
...

We see that the first term, , is -2 and we want to find
the sum of 18 terms, so n is 18
...

We can see that each term increases by

from one to

the next, so plug in for d
...

18 minus 1 is 17
...


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Then, add -2 and 34
...
Now that we know the
value of the 18th term, we can use the formula for
finding the sum
...
We know that
n = 18
...

-2 plus 32 is 30
...

18 divided by 2 is 9
...

9 times 30 is 270
...


CONCEPT
Finding the Sum of an Arithmetic Sequence

22

It took Todd 11 hours to travel over pack ice from one town in the
Arctic to another town 330 miles away
...
Assume the pack ice was drifting at a
constant rate, and that Todd’s snowmobile was traveling at a
constant speed relative to the pack ice
...
com)

•

10 miles per hour
•

4 miles per hour
•

26 miles per hour correct
•

20 miles per hour
RATIONALE
It took Todd

hours to travel over pack ice from

one town to another town
miles away
...
During his return journey, it
took him
hours to go the same distance
...
Two rates are
involved in this problem: the rate of Todd’s
snowmobile and the rate of the drift of the pack
ice
...
Next, consider the
combined speed when the rates work together
and when they work against each other
...
When two rates work against each other,
the opposing rate is negative, which produces an
expression involving subtraction
...

Both equations must be written in the form of
, which states that distance is equal to

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On the trip back, the distance, , is
miles, the combined rate is

, and the

time, , is
hours
...

In the first equation, divide both sides by
which equals

,

on the left side and leaves

on the right side
...
To
solve for the rates, add the two equations
...


The terms in each equation add to
and
is eliminated
...

Dividing
by equals

...


CONCEPT

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Find the solutions to the equation by using the quadratic formula
...
However, be sure that the equation
is set equal to zero
...


from

Now that the equation is set equal to
zero, identify the coefficients , , and
that are found in the standard equation,

...
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In the equation
and

,

,

,


...

Plugging the values of , , and into the
quadratic formula will produce the solutions
to the quadratic equation
...

First, simplify what is underneath the
radical, or


...
Next, find the difference
and


...
Next, take


...
Next, evaluate

the denominator by multiplying

and
...

One equation involves subtraction, while
the other equation involves addition
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To
find the solutions, divide the numerators by
the denominator,
...


CONCEPT
The Quadratic Formula

24

What is the solution set for the following inequality?
x + 4 < 3(x – 2)
•

x > 5 correct
•

x>1
•

x<1
•

x<5
RATIONALE

Before solving for x, make sure that each side of the
inequality is fully simplified
...

On the right side, 3(x – 2) is equivalent to 3x – 6
...
First, subtract 3x from both
sides
...
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Subtracting 3x from both sides cancels the 3x term on the
right side of the inequality, leaving x terms on only the left
side
...

Subtracting 4 from both sides leaves only the x term on the
left side
...
Remember that the sign of the inequality sign
changes whenever you multiply or divide by a negative
number!
The solution to this inequality is x > 5
...

•

•

correct
•

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Here, we will use FOIL to multiply
, but you can choose any two
binomials to start
...

Once the binomials have been multiplied together,
evaluate the multiplication
...
The

next step is to combine like terms,
and

combined is

expressed as

and


...


can be


...


When multiplying these two terms, we will start by
distributing

into


...


times
equals

...
We will distribute
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times
equals

...
The final
step is to add these two parts together
...


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