College Algebra Classnotes:A companion to the ALEKS system by McGraw-Hill

Prepared by Richard G. Ligo

Dear reader,

Thank you for using this resource! These classnotes strive to be a source of assistancefor both instructors and students, bringing focus and emphasis to key ideas necessary forfurthering one’s education in mathematics, algebra in particular.

To instructors: You will hopefully find that these notes have the ability to streamline yourteaching. Traditional lecture time can be in short supply in ALEKS-based courses, butthese notes attempt to bring succinct focus to the most essential examples. If you have aneasy method by which to disseminate completed copies of these notes to students, I highlyrecommend that one do so. I have found that such tactics have the ability to significantlyminimize problems caused by student absences.

To students: I’m happy that you have decided to step into the world of algebra! While itmay not seem like the most thrilling topic in the world, I encourage you to find the beautyburied in the structure and organization of the subject. With regard to these notes, I wouldhope that you would find them both straightforward and enlightening. These notes are notintended to replace the explanations given by ALEKS, but rather, provide an additional(sometimes simpler) perspective. Good luck!

Best,Rich

Disclaimer: The problems contained within these notes are both of my creation, as well asthat of McGraw-Hill. I also do not claim that these notes (or my accompanying solutions)are entirely free of typographical errors.

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Module 1 Name:

Module 1 contains a wide array of topics, but most can probably be considered review. Whilethese topics are fairly straightforward, solid knowledge of them is critical to your success inalgebra.

First, we revisit the order of operations. Just to reiterate, that order is the following:

1)

2)

3)

4)

Use the order of operations to evaluate the following examples:

Ex 1: 42[(13 + 4)− 8] Ex 2: 3[4 + 3(4 + 1)]

Sometimes we begin with an expression containing variables, then evaluate it by substitutingin a value for the variable. Do this to the following examples:

Ex 3: x2 + 3x− 5, x = −2 Ex 4:−3y − 1

y2 − 3y + 5, y = 3

The distributive property is also quite valuable. Use it to simplify the following examples:

Ex 5: 8(x + 5) Ex 6: −13

(21− 15x)

3

Much of our knowledge culminates in the simplification of expressions. Simplify the following:

Ex 7: −5(5y − 9) + 3(3y + 6)

Ex 9: −10 + x + 4x− 7− 4x

Ex 11:(2xy−3)−2

(3x−2y4)−3

Ex 13: (x + 10)(x− 10)

Ex 15: (2x + 3)(2x2 − 5x− 7)

Ex 8: 16x− 1

8x + 1

2x2

Ex 10: 2x + 3y − 5x + 7z

Ex 12:(3−1x−3y)−1(2x2y−3)2

(5x−2y2)−2

Ex 14: (−x + 3)(2x− 6)

Ex 16: (7x + 2y)2

4

Module 2 Name:

Factoring constitutes one of the pillars of algebra, so it is necessary for us to revist the subject.

In this class, have three types of factoring at our disposal:

1)

2)

3)

Of the special factoring types, we have three forms:

Use the tools listed above to factor the following examples, note that some may require morethan one strategy:

Ex 1: 8x2 + 6x

Ex 3: 9x10 − 49

Ex 5: x3 + 8

Ex 2: x2 − 2x− 8

Ex 4: 36x6y + 45x5y4 + 81x3y2

Ex 6: 24x4 + 17x2 − 20

5

It is also crucial that we have a solid understanding of radicals and rational exponents.Change the following examples between radical and exponential form:

Ex 7: x54 Ex 8: ( 7

√x)9 Ex 9:

√x2y3z5

Now simplify the following expressions, rationalizing as necessary:

Ex 10: 4x√

63y + 6√

28x2y

Ex 12: (2√

7 + 3)2

Ex 14:

(−125x9

27y6

) 23

Ex 16: ( 3√

5− 3√

4)( 3√

25 + 3√

20 + 3√

16)

Ex 11:3

2 +√

5

Ex 13:

√9

7

Ex 15: (2x−14 y

34 )4

Ex 17: −2 3√−125x3y7

6

Module 3 Name:

Perform the indicated operations, simplifying as necessary.

Ex 1:5x + 20

3x + 12

Ex 3:2x2 − x− 1

2x2 + 5x + 3÷ 4x2 − 1

2x2 + x− 3

Ex 2:2(x + y)

y· 18

6(x + y)2

Ex 4:2x2 − 3x− 5

2x2 − 7x + 5

Recall the process for determining the least common denominator:

Ex 5:1

9x+

5

12x2Ex 6:

x + 3

x2 − 9− −1

x2 + x− 12

7

Simplifying the following complex fractions:

Ex 7:

xpq3

p2

qx2

Ex 8:35m− 2

m2

92m

+ 34m2

Solve the following linear equations for x:

Ex 9: 2x + 9 = 4x + 11

Ex 11: 11x− 5(x + 2) = 6x + 5

Ex 13: 5x + 8 = 7 + 3x

Ex 10: 11x− 5(x + 2) = 6x + 5x

Ex 12: 3(2x− 4) = 6(x− 2)

Ex 14: 12x− 5 = 11x + 5− x

Ex 15: If 4 is multiplied by a number decreased by 7, the product is 100. Find the number.

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Module 4 Name:

While most of our work thus far has focused on expressions and equations, it’s time for us toturn our attention to inequalities. There are actually four ways for us to write an inequality;give an example of this below:

Write each example in the four forms of an inequality:

Ex 1: x ≤ 3

Ex 3: (−2, 5]

Ex 2: (−∞,−1) ∪ [12,∞)

Ex 4: {x | 2 ≤ x}

Sometimes an inequality must be solved first. Solve the following examples for x and thenrepresent each inequality in all four ways:

Ex 5: 5x− 11 ≥ 2x− 5 Ex 6: −6(x− 1) + 2x ≤ −2(2x− 3)

9

Recall the definition of the absolute value:

We can then solve equations involving absolute values:

Ex 7: 2|x− 1| − 7 = 3

Ex 9: −|x + 3| − 14 = −14

Ex 8: −|3x + 4|+ 3 = −5

Ex 10: |2x− 1| = |x + 3|

Finally, we can combine inequalities and absolute values. Solve the following inequalitiesand graph the solutions:

Ex 11: 3|x + 4|+ 5 < 8

Ex 13: |x + 3| > 7

Ex 12: −2|x|+ 3 > 7

Ex 14: 4|5− 2x| − 9 ≤ −9

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Module 5 Name:

Recall the definition of the imaginary unit, as well as some of its properties:

The gives us the definition of a complex number:

Simplify the following complex expressions:

Ex 1:√−144

Ex 4:√−4975

Ex 7: 6+√−273

Ex 2: (12−√−4)−(7−

√−9)

Ex 5: (−2 + 5i) + (3− i)

Ex 8: 5i(−3i)

Ex 3: (−4− 2i)(3 + 2i)

Ex 6: (2− i)2

Ex 9: 21−√−4

Complex number arise as solutions to quadratic equations. Recall the standard form of aquadratic equation:

We have four methods for solving quadratic equations:

Solve the following quadratic equations by factoring:

Ex 10: x2 − 2x− 15 = 0 Ex 11: −3x(x + 4) = −36

11

Solve the following quadratic equations by using the square root property:

Ex 12: x2 = 16 Ex 13: (x + 5)2 − 9 = 40

Solve the following quadratic equation by completing the square:

Ex 14: x2 + 6x = −5

Recall the quadratic formula:

Solve the following quadratic equation by using the quadratic formula:

Ex 15: 2x2 − 4x = −5

What do we know about the solutions of a quadratic before simplifying the formula?

Write a quadratic equation satisfying the following conditions:

Ex 16: Roots at x = 3 and x = −2 Ex 17: Roots at x = 1 and x = 7, and aleading coefficient of 3

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Module 6 Name:

The rational and radical expressions from Module 4 now return in equations! Solve the fol-lowing rational and radical equations, and don’t forget to check your answers.

Ex 1: x3

+ 12

= 16

Ex 3: 2x− 1

x−1 = 4x2−x

Ex 5: 2 3√x− 5 + 4 = 0

Ex 2: x + 12x−3 = 1 + 4x

x−3

Ex 4: x2−8x−3 = 1

x−3

Ex 6:√x + 15−

√x + 3 = 2

13

We can also use some of our tactics from Module 5 to solve special forms of polynomialequations. Solve the following equations.

Ex 7: 4x3 − 40x = 6x2 Ex 8: x4 − 36 = 5x2

We also make a return to graphing. Graph the following equations on the axes provided byusing a t-table.

Ex 9: y = x2 + 2x

Ex 10: y = −x3 + 1

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Module 6 (continued) Name:

When we talk about graphing, there are two important formulas we must keep in mind:

Determine the find the midpoint of and the distance between the following ordered pairs:

Ex 11: (−2,−2) and (4, 6)

The tools we have learned up to this point now allow us to graph circles! The standard formfor the equation of a circle is as follows:

Graph the circles represented by the following equations:

Ex 12: (x + 1)2 + (y − 1)2 = 4 Ex 13: (x− 2)2 + (y + 3)2 = 12

15

Occasionally our circle’s equation will require us to complete the square to get it into standardform. Determine the standard form of the following equation and graph the circle.

Ex 14: x2 + y2 − 10x + 4y + 4 = 0

Given a set of information, we can also write the equation corresponding to that circle. Foreach of the following situations, write the equation of the corresponding circle.

Ex 15: Center at (4,−3), radius of 9

Ex 16: Diameter has endpoints (5, 1) and (5, 7)

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Module 7 Name:

Today’s work focuses entirely on the graphing of lines. First, we must talk about the ideaof slope. Sketch examples of lines with positive, negative, zero, and undefined slope.

There are a variety of ways we can mathematically interpret the slope:

There are three types of equations we frequently use to represent lines:

Graph the following lines using their equations:

Ex 1: y = 2x− 3

Ex 3: x = 2

Ex 2: y − 3 = −3(x + 1)

Ex 4: 2x− 6y = 18

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All the concepts on the previous page culminate in the task of writing an equation for a linesatisfying some given set of information. Write an equation for the line described in eachexample, and then graph the line.

Ex 5: The line through the points (1, 1) and (−2,−5)

Ex 6: The line through the point (2,−4) and parallel to the line y = 75x + 5

Ex 7: The line with a y-intercept of 4 and perpendicular to the line 3x + 2y = 5

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Module 8 Name:

This conceptually-heavy lesson consists mainly of definitions and pictures, with symbolic al-gebra frequently playing a smaller role. First we need to talk about functions. Which of thefollowing represents a function?

4

3

2

1

8

7

6

5

4

3

2

1

8

7

6

5

f(x) = x2

From the idea of a function we get the concepts of domain and range:

Write the domain and range of the functions above:

We can combine the concepts of domain and range with our knowledge of slopes and inter-cepts to more completely describe the graph of a function.

For each of the following graphs, state the domain, range, x and y intercepts, and intervalsof increase and decrease.

Examining the slope allows us to talk about local extrema. We define a local minimum andlocal maximum as follows:

Now determine the local minima and local maxima of the above functions, as well as thevalues at which these minima and maxima occur.

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We can frequently determine the domain of a function simply by considering its algebraicrepresentation. In particular, we will consider radical and rational functions, so determinethe domains of the following functions:

f(x) = 2x2+5x−32x+4

f(x) =√

3x− 6 f(x) = x100−21x11√x2−1

A function can sometimes be said to be even or odd. We define these concepts as follows:

These definitions allow us to consider functions both graphically and algebraically. Determineif the following functions are even, odd, or neither:

f(x) = 12x2 − 4 f(x) = x3 − 4x f(x) = 1

2x + 1 f(x) = −x5 + x3 + x

Once we understand the graph of a function, we can change it by translating, reflecting, orscaling it. Graph the following examples without using a table.

f(x) = −x2 + 2 f(x) = |x| − 5 f(x) = (x− 3)2 − 2

For parabolas, we can define the following:

Find the vertices, axes of symmetry, and x intercepts for the parabolas above.

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Module 9 Name:

Module 9 is a continuation of our journey into the world of functions and graphing. Firstwe talk about a specific class of functions, called piece-wise functions.

In each of the examples, graph the function and evaluate it at x = −3, x = 0, and x = 2

Ex 1: f(x) =

{−x− 2 x < 0

x2 x ≥ 0Ex 2: g(x) =

{3 x ≤ 213x + 1 x > 2

As with real numbers, we can define addition, subtraction, multiplication, and division offunctions. Let f(x) = x2 − 5x and g(x) = 2x− 9 and determine the following:

Ex 3: (f + g)(x) Ex 4: (f − g)(x) Ex 5: (f · g)(x) Ex 6: fg(x)

We can also define another type of operation, called composition:

Let f(x) =√x + 4, g(x) = 3x + 2; find the following compositions and state their domains.

Ex 7: (f ◦ g)(x) Ex 8: (g ◦ f)(x)

A somewhat trickier idea reverses function composition. Write each of the following functionsas the composition of two (non-identity) functions. (There are multiple answers.)

Ex 9: h(x) = ( 3√x + 1)2 − 3 Ex 10: k(x) = 4(x3 + 7)8

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Function notation also gives us a better way to deal with translations of a graph. Write theequation of the following translations, then graph it along with the original function.

Ex 11: f(x) = x2, up 2, right 3 Ex 12: g(x) = |x|, down 5, left 1

We now return to the world of parabolas, this time with some new tactics. The vertex formof a parabola is as follows:

Graph the following parabolas, determine the minimum/maximum, and write the first ex-ample in vertex form.

Ex 13: y = x2 − 6x + 5

Ex 14: y = 2x2 + 8x + 3

It is also possible to go from the graph of a parabola to its equation!

Ex 15: Write the equation corresponding to the following graph

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Module 10 Name:

Module 10 is all about polynomials and their roots. Write a polynomial satisfying the fol-lowing conditions:

Ex 1: Find a polynomial of degree 5 with zeros at 4 (multiplicity 2), 2, 0, and −1.

We now must state two theorems:

We will be using these facts over the course of Module 10.

We will be using these facts in tandem with an old skill: polynomial long division.

Ex 2: x3 − 4x2 + x + 6÷ x− 1 Ex 3: 2x3 + 3x + 7÷ x− 3

Factor the following trinomial, given a specific zero.

Ex 4: x3 + 2x2 − 5x− 6 with zero at −3

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We can use some of these skills to analyze rational functions. Find the horizontal and verticalasymptotes of the following rational functions.

Ex 5: f(x) = 3x2+12x+12x2−1 Ex 6: f(x) = 5x+2

x2+2

The following rules determine the locations of the asymptotes:

Module 10 also revisits inequalities, this time with quadratic and rational terms. Solve thefollowing inequalities.

Ex 7: x3 + x2 < 12x

Ex 8: 6x−6 ≥ 3

Ex 9: x−1x+2

> x−5x−3

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Module 11 Name:

We begin with a new definition:

Determine if the following functions are one-to-one.

A function that is one-to-one is also .

Say that we believe that g(x) and f(x) are inverse functions. We can confirm this by verifying

Determine if the following pairs of functions are inverses:

Ex 1: f(x) = 6x + 1, g(x) = 6x− 1 Ex 2: f(x) = 4x, g(x) = 4

x

We also have a method for finding the inverse of a function. Find the inverses of the functionsin the following examples and state their domain and range:

Ex 3: f(x) = x−135

Ex 5: f(x) =√x− 1 + 1

Ex 4: f(x) = 6x7x−1

Ex 6: f(x) = 3− x3

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A fundamental tool in statistics and growth modeling is the idea of an exponential function.Graph the following exponential function by using a table of values.

Ex 7: f(x) = 2x + 1

Combining the graph of an exponential function with our knowledge about translations allowsus to graph other exponential functions with considerably less work. Graph the followingexponential functions, state their domain and range, and graph any asymptotes.

Ex 8: f(x) = 4x−1 + 2 Ex 9: f(x) = −(13)x − 1

Ex 10: A city has a population of 310,000 people. Suppose that each year the populationgrows by 5.25%. What will the population be after 11 years?

Solve the following equations by finding common bases:

Ex 11: 16x+3 = 32 Ex 12: 364−2x = 6−6x

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Module 12 Name:

Our discussions of exponential and inverse functions very naturally flow toward the conceptof a logarithmic function. We can define as logarithmic function as the following:

This definition allows us to convert between logarithmic and exponential notation.

Ex 1: log515

= −1 Ex 2: log2 16 = 4 Ex 3: ln 5 = x

Logarithms were not simply created to provide an inverse for exponential functions. Thefollowing properties made them extremely valuable to 17-19th century scientists:

It should be noted that these properties are also treasured by modern mathematicians.

While logarithms of most numbers are now simply approximated by a calculator, combiningtechniques from earlier this year with these properties allow us to determine the exact valuein certain cases.

Ex 4: log6 1 Ex 5: log3 27 Ex 6: log218

Now expand or collapse the following logarithmic expressions using the properties above:

Ex 7: log(xy6)

Ex 9: log(√xz4

y5)

Ex 8: 15

log8 x + 6 log8 y − log8 x

Ex 10: 2 logm x + 3(logm y − 5 logm z)

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We can also use these properties to help us solve logarithmic equations:

Ex 11: −2 log2(−4x) = −6 Ex 12: 5 + log3(2x) = 6

We can also graph logarithmic functions, although this can require careful choices for x.Graph the following function using a table and state the domain and range.

Ex 13: f(x) = log2 x

We can combine this knowledge with ideas about translations to graph other logarithmicfunctions. Graph the following logarithmic functions and state their domain and range.

Ex 14: −2 + log1/2 x Ex 15: f(x) = −3− log3 x

Finally, we can use our knowledge of inequalities to determine the domain of particularlycomplicated rational functions:

Ex 16: f(x) = ln(9− x2)

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Module 13 Name:

Module 13 ventures a bit deeper into exponential and logarithmic functions, and we see theseconcepts applied to some real-world situations.

First we learn some additional techniques for solving logarithmic and exponential equations.If we want our answer to be exact (rather than rounded), it often must be left in terms of alogarithm. Be sure to check your answers to logarithmic equations.

Ex 1: ln 20 = ln(x + 2) + ln 5

Ex 3: log5(x− 6) = 1 + log5(x + 2)

Ex 5: 22x − 9(2x) + 8 = 0

Ex 2: 137x = 2x−2

Ex 4: e2x − 2ex − 35 = 0

Ex 6: − log2(x + 8) = 4− log2(x− 7)

In those cases when an approximate answer is good enough, we can simply use a calculatorto evaluate the logs left over at the end of the problem. This is also true when consideringat exponential expressions.

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There are two formulas critical to our success in Module 13:

Apply the previously stated formulas to the problems below:

Ex 7: Rich’s brother Joe wants to buy a Tesla Model S, so he takes out a loan of $9000 atan interest rate 18% of compounded each year. How much does Joe owe in a one year? Howmuch does he owe at the end of two years?

Ex 8: Rich, being the older and wiser brother, decides to invest his money. He invests $8000at a rate of 7.5% in a certificate that is compounded semiannually. How much does he haveafter seven years?

Ex 9: The number of people at a certain fraternity party increases according to a continuousexponential growth model, with a growth rate parameter of 68% per hour. The party startedwith just a small group, and after four hours, the party has grown to 364 people. Find thenumber of people present when the party started.

Ex 10: Rich’s father, being an accountant, knows that investing in a continuously com-pounded fund is the best way to get a solid return. He invests $2600 in a particular accountat a fixed interest rate, compounded continuously, and after three years his money has grownto $2824. Find the interest rate per year.

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Module 14 Name:

Module 14 is filled with word problems, but first we must revisit and extend an old skill.

Systems of equations are sets of equations with several variables. There are three types ofsolutions sets for systems of equations:

The best way to visualize these ideas is through graphing. State whether each graph repre-sents a consistent independent, consistent dependent, or inconsistent system.

Ex 1: y = x+ 2, y = x+ 3 Ex 2: y = −2x+ 1, y = −4( 12x− 1

4) Ex 3: y = 3

5x− 5, y = −5x+ 4

Solve the following systems using graphing:

Ex 4: −x− 3y = 3, y = −13x + 1 Ex 5: 2x + y = −5, y = 1

3x + 2

We can also solve via substitution or elimination:

Ex 6: x = 3y − 4, 3x + 5y = 30

Ex 8: 4x + 9y = −14, −3x− 2y = 20

Ex 7: x + 2y = 4, −2y = x− 4

Ex 9: x + 2y + 6 = 0, −x− 2y = −6

31

We can extend these ideas to systems of three equations with three variables.

Ex 10: −x + 2y + 2z = 7, 2x− y − 2z = −6, 3x− 2y − 4z = −9

Now use these ideas to solve the following word problems:

Ex 11: The sum of two numbers is 48. One number is times as 3 large as the other. Whatare the numbers?

Ex 12: Rich and Joe are unloading hay (again). Rich worked for 6 hours and Joe workedfor 4 hours. Together they unloaded 2360 bales of hay, and their rates combined is 440 balesper hour. How fast is each one?

Ex 13: Rich leaves from Iowa City at the same time Joe leaves from Pennsylvania (660 milesaway), and they drive towards each other on I-80 at a constant speed. If Rich drives 45 mphfaster than Joe and they meet after 4 hours, what were their individual speeds?

Ex 14: Rich, Joe, and Laura (Joe’s girlfriend) sent 711 texts last week. Joe sent 10 timesmore than Rich, and Laura sent 60 more than Joe. How many did they each send?

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