Algebra/Chapter 2/Exercises

A set of exercises related to concepts from Chapter 2.

This set contains 73 problems (8 Conceptual Questions + 64 Exercises + 1 Project)

Conceptual Questions

Q2.1 (Are they the Same?) Is “3 less than a number” the same thing as “the difference of 3 and a number”? What about "3 more than a number" and "the sum of 3 and a number"? Explain your reasoning in both cases.

Q2.2 (Coefficients) Does the expression x + 2 have any coefficients for x? What about the expression yx + 2, where any number can take the place of y? If so, identify the coefficient of x in each expression? If not, explain your reasoning.

Q2.3 (Zero as a Constant Term?) An expression like x + y can be rewritten as x + y + 0. If this is the case, is it necessarily true that zero is a constant term for any given expression?

Q2.4 (Grammar in Mathematics) If mathematical expressions are analogous to "nouns" in a language, what part(s) of a mathematical expression is/are analogous to "verbs"? What part(s) are analogous to "conjunctions"?

Q2.5 (Set of Sets) Give three examples of sets whose elements are sets.

Q2.6 (Set of Sets of Sets) Give an example of a set whose elements are sets of sets.

Q2.7 (Multiplication and the Distributive Law) Point out in what sense the usual arrangement of the multiplication of 365 and 392 is an instance of the Distributive Law.

Q2.8 (Box of Blocks) Amanda and Billy stack a pile of blocks to build a box pictured on the left, and would like to figure out how many blocks they used. Since the box is solid, each layer has the same number of blocks as the top layer. Amanda states that there are 4 x 5 = 20 blocks on the top, and since there are 4 layers below it, she says there are 20 x 5 blocks total. Who is correct?

Exercises

Section 2.1

(★) 2.1 (Writing/Simplifying Expressions) Write an expression that best represents the following.

1. The sum of a and 3.

(★) 2.2 (Evaluating Expressions) Evaluate each expression for the given variable value.

$1.\ x+3,\ x=3$	$2.\ 10x-3,\ x=3$	$3.\ 4a-2,\ a=9$	$4.\ {\frac {-n-1}{3}},\ n=11$	$5.\ {\frac {-b-2}{7}},\ b=5$
$6.\$	$7.\$	$8.\$	$9.\$	$10.\$
$11.\$	$12.\$	$13.\$	$14.\$	$15.\$
$16.\$	$17.\$	$18.\$	$19.\$	$20.\$

(★) 2.3 (Anatomy of a Mathematical Expression) Look at the expression below. Define all of the simplified expression's terms, variables, coefficents, and constants.

$3x^{2}+4y+y+6$

(★) 2.4 (Constants and Variables) Letters will be given to represent various numbers. Decide if the following quantities should be referred to as variables or constants.

$a.\ T$, the temperature outside of your house.
$b.\ F$, the number of fingers on an average person's hand.
$c.\ P$, the price of a gallon of gas.
$d.\ L$, the number of leaves on a tree.
$e.\ S$, the number of sides on a rectangle.
$f.\ I$, the number of inches in a foot.
$g.\ Y$, the number of years since the last moon landing.
$h.\ D$, the number of donuts in an unopened box of a dozen.
$i.\ W$, the number of windows open on your computer screen.
$j.\ R$, the number of problems in this book you have attempted.

Section 2.2

(★) 2.5 (Writing Mathematical Sentences) Write a sentence that best represents the following.

Section 2.3

(★) 2.6 (Identifying Mathematical Statements) Which of the following sentences are statements?

1. Eat your vegetables!
2. Aaron ate his vegetables.
3. Every rectangle is a parallelogram.

Section 2.5

(★★) *2.7 (Square Root of 3) Prove that ${\sqrt {3}}$ is an irrational number.

(★★) *2.8 (Chess Board I)

(★★) *2.9 (Chess Board II)

(★★★) *2.10 (Density Property of Real Numbers) The Density Property of Real Numbers states that between any two real numbers, there is another real number. Use this property to prove that there are infinitely many real numbers between 0 and 1.

Section 2.6

(★) 2.11 (Element or Not?) Determine if the number 10 is an element in the following sets.

1. $\{4,5,6,...,15\}$
2. $\{1,2,3,4,5,...\}$
3. $\{1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},...\}$
4. $\{x|x\ is\ a\ whole\ number\ greater\ than\ 11\}$
5. $\{x|x\ is\ a\ whole\ even\ number\}$
6. The set $C$ made up of composite numbers

(★) 2.12 (Roster Notation) Each of the sets below are defined using roster notation.

i. $\{1,4,9,16,25...\}$
ii. $\{0,4,8,...,96,100\}$
iii. $\{3,9,15,21,27...\}$
iv. $\{...,-\pi ^{4},-\pi ^{3},-\pi ^{2},-\pi ,1\}$

1. Determine four other elements that may appear in the sets above.
2. Use set builder notation to describe the sets above.

(★) 2.13 (Sorting Automobiles) Construct a Venn Diagram which illustrates the possible unions and intersections of the following sets relative to the universal set consisting of automobiles made in the United States.

$F:Fourdoor,S:Sunroof,P:Powersteering$

(★) 2.14 (Relating Types of Numbers) Refer to the section Types of Numbers in Section 1.8. Create a Venn Diagram which shows how each of the number types listed are related to each other.

(★) 2.15 (Working with Sets I) Let $U=\{0,1,2,3,4,5,6,7,8,9,10,11,12,13\}$, $M=\{0,2,4,6,8\}$, $N=\{1,3,5,7,9,11,13\}$, $Q=\{0,2,4,6,8,10,12\}$, and $R=\{0,1,2,3,4\}$.

Use these sets to find the following.

(★) 2.16 (Working with Sets II) Let $U=\{copper,sodium,nitrogen,potassium,uranium,oxygen,zinc\}$, $A=\{copper,sodium,zinc\}$, $B=\{sodium,nitrogen,potassium\}$, $C=\{oxygen\}$.

Use these sets to find the following.

(★★) 2.17 (Working with Sets III) If $U=\{x|0<x<12\}$, $M=\{x|1<x<9\}$, and $N=\{x|0<x<5\}$.

Use these sets to find the following.

(★★) 2.18 (Working with Sets IV) Suppose $A$, $B$, and $C$ are subsets of the universal set $U$.

Using Venn Diagrams, shade the areas that represent the following.

(★) 2.19 (Working with Subscripts) For a whole number $j$, $x_{j}=(-1)^{j}$. Find the value of $x_{0}$, $x_{1}$, and $x_{183}$.

(★) 2.20 (List of Numbers) Refer to the set of numbers below, and use it to answer the following questions.

$x=\{5,11,14,9,3,25,16,8,1,11,68,63,43,99,35,100\}$

1. In the set, which number is represented by $x_{3}$?
2. In the set, which number is represented by $x_{10}$?
3. What symbol(s) can be used to represent the number 25 in the list?
4. What symbol(s) can be used to represent the number 11 in the list?
5. What number represents $n$ in $x_{n}$?
6. What value does $x_{n}$ take?

2.21 (Average) Use subscript notation to write an expression which represents the average of $n$ numbers.

Section 2.7

(★★) 2.22 (Gauss's Trick) In the late 1700s, the kindergarten class of the mathematician Carl Fredrich Gauss was asked to find the sum of all of the natural numbers between 1 and 100. While most of the class had struggled with this seemingly impossible task, Gauss was able to determine the solution to this problem rather quickly.

1. How was he able to do this?

2. We can use techniques similar to Gauss' in order to find the sum of several numbers. Can you find the following?

i. $1+2+3+4+...+201$
ii. $2+4+6+8+...+200$
iii. $101+102+103+...+998+999+1000$
iv. $9+12+15+...+54+57+60$

(★) 2.23 (Using the Distributive Property) Use the Distributive Property to simplify these expressions.

$a.\ 2(14x-26)$
$b.\ (2/3)(3x+9)$
$c.\ 3(12x+4y)$
$d.\ 2(5x-6)+3(3x+2)$
$e.\ (4x+7)(2x-3)$
$f.\ (x+1)(x+2)(x+3)$

(★★) 2.24 (Distribution with Three Terms) What is the coefficient of y in the expansion of the expression below?

$(5x+2y-4)(2x+7y+3)$

(★) 2.25 (Closure of a Set) Two letters from the set $\{a,b,c,d,e\}$ are chosen and multiplied together. The results after doing this are as follows:

*	a	b	c	d	e
a	b	c	e	a	d
b	d	a	c	b	e
c	c	d	b	e	a
d	a	e	d	c	b
e	e	b	a	d	c

Is the set closed under multiplication?

(★) 2.26 (Closure of Operators) Complete the following table which represents the closure properties of operations with different types of numbers. Use a check mark to represent closure, and a cross to represent no closure.

	Addition	Subtraction	Multiplcation	Division	Exponentiation	Root
ℕ
𝕎
ℤ
ℚ
𝕀
ℝ

2.27 (Determining Properties of Real Numbers) Determine if the following statements are always, sometimes, or never true. If the statement is always true, explain your reasoning. If the statement is not always true, provide a counterexample.

$a.\ An\ integer\ is\ a\ whole\ number.$
$b.\ If\ a\ number\ is\ whole\ it\ is\ a\ natural\ number.$
$c.\ If\ a\ number\ contains\ a\ decimal\ it\ is\ an\ integer.$
$d.\ If\ a\ number\ is\ nautural,\ then\ it\ is\ a\ real\ number.$
$e.\ The\ product\ of\ two\ irrational\ numbers\ is\ an\ irrational\ number.$

2.28 (Identifying Properties of Real Numbers) Identify the following properties being expressed.

$a.\ 4(3x+4)=12x+16$
$b.\ 6+0=6$
$c.\ (2+7)+5=(2+5)+7$
$d.\ (3/4)(4/3)=1$
$e.\ To\ divide\ 3072\ by\ 512,\ you\ can\ divide\ 3072\ by\ 16,\ again\ by\ 8,\ and\ again\ by\ 4.$

2.29 (Product Pattern) Use the Associative Law to explain why the products in each rule are equal.

$2*2=1*4$

$4*3=2*6$

$6*4=3*8$

$8*5=4*10$

$10*6=5*12$

$12*7=6*14$

$14*8=7*16$

2.30 (Suare of Sum/Difference) For two numbers $a$ and $b$, find the following:

$a.\ (a+b)^{2}$
$b.\ (a-b)^{2}$

2.31 (Secret of 1001) A boy claims that he can figure out the product of any three digit number and 1001. A student in his arithmetic class challenges him to find the product of 1001 and 865, and he gets the correct answer immediately. Compute the answer, and determine the boy's secret.

Section 2.10

(★) 2.32 (Museum Admissions) The total cost to get admission to a museum is 25a + 10c + 8s for a adults, c children, and s seniors. How much does it cost for each adult, child, and senior respectively to get admission? If a family of two adults, three children, and one senior wants to gain admission to the museum, how much would they have to pay in total?

(★) 2.33 (Cutting Edge) A 12 ft long piece of rope was cut into two pieces of different lengths. Use one variable to represent the lengths of the two pieces.

(★) 2.34 (Play Ball!) The diameter of a basketball is approximately 4 times of that of a baseball. Express the diameter of a basketball in terms of the diameter of a baseball.

(★) 2.35 (Pocket Change) Suppose have d dimes and n nickels in your pocket. Write an expression which represents the total amount of money you have. Use this expression to figure out how much money you would have if you had 9 dimes and 7 nickels.

(★) 2.36 (Units of Temperature I) The formula

$C={\frac {5}{9}}(F-32)$

expresses the relationship between Farenheit temperature, F, and Celcius temperature, C. Use this equation to convert $50^{\circ }F$, $86^{\circ }F$, $32^{\circ }F$, $100^{\circ }F$, and $-50^{\circ }F$ to their equivalent temperature on the Celcius scale.

(★) 2.37 (Units of Temperature II) The formula

$K=C+273.15$

expresses the relationship between Celcius temperature, C, and Kelvin temperature, K. Use this equation to convert $-273.15^{\circ }C$, $30^{\circ }C$, and $100^{\circ }F$ to their equivalent temperature on the Kelvin scale.

(★★) 2.38 (Chemical Formula for Sugar) The chemical formula for glucose (sugar) is $C_{6}H_{12}O_{6}$. This formula means there are 12 hydrogen atoms for every 6 carnon atoms and 6 oxygen atoms in each molecule of glucose. If x represents the number of atoms in oxygen in a pound of sugar, express the number of hydrogen atoms in the the same pound of sugar.

(★) 2.39 (Building Blocks) Look at the arrangements of building blocks below. How many blocks will appear in diagram 17?

(★) 2.40 (Rows of Seats) In an auditorium, the first row has 8 seats, and five extra seats are added in each row afterwards. If the auditorium has 20 rows, how many seats are there in total?

(★★) 2.41 (Triangles in Polygons) In a triangle, there are three sides. We can obviously observe from this that this contains 1 triangle. In a quadrilateral, there are four sides. We can observe from this that two non-overlapping triangles can be made out of this by dividing it along its corners. In a pentagon, there are five sides. We can observe from this that three non-overlapping trianges can be made out of this by dividing it along its corners. Using this information, how many non-overlapping triangles can you make out of a decagon (10-sided polygon) by dividing it along its corners?

(★★) 2.42 (Product of Consecutive Numbers) Two numbers are consecutive if they follow each other in numerical order. For example, the numbers 4 and 5 are consecutive because 5 comes after 4. What would be an algebraic representation of the product of two such numbers?

(★★) 2.43 (Odd Numbers) Write an expression that represents the nth odd number, O. (First odd number is 1, Second odd number is 3, and so forth) Afterwards, use this expression to find the 143rd odd number.

(★★) 2.44 (Weight-Loss Points) Several weight-loss programs assign points to prepared or packaged foods that take into account of the food's fat F, carbohydrate C, protein P, and fiber B content in grams. The point value for a given food item can be represented by the following expression:

$W={\frac {F}{4}}+{\frac {C}{9}}+{\frac {P}{10}}-{\frac {B}{12}}$

Determine the point value of one serving of the item having the nutrition facts on the left.

(★★) 2.45 (Adjusted Poverty Threshold) The adjusted poverty threshold for a single person between 1999 and 2013 can be approximated by the formula

$y=2.719x^{2}+196.1x+8718$

where x is the number of years since 1999, and where y is the average adjusted poverty threshold. According to the model, what was the average adjusted poverty threshold in 2005? In 2012?

(★★) 2.46 (Period of a Pendulum) The period t, in seconds, of the swing of a pendulum is given by

$t=2\pi {\sqrt {\frac {L}{32}}}$

where L is the length of the pendulum in feet. Find the period of a pendulum 8 feet long.

(★★) 2.47 (Velocity of a Ball) A ball is thrown vertically upward. Its velocity t seconds after its release is given by the formula:

$v=v_{0}-gt$

where v0 is its initial velocity, g is the acceleration due to gravity, and v is the velocity of the ball at time t.

1. Calculate the upward velocity of a ball after 8 seconds on Earth (g = 9.8) after tossing it at a velocity of 5 m/s.
2. Calculate the upward velocity of a ball after 8 seconds on the Moon (g = 1.625) after tossing it at a velocity of 5 m/s.

(★★) 2.48 (Isotopes) The isotopes of any given chemical element all have the same atomic number (same number of protons), but a differing numbers of neutrons. They are written as the formula:

$_{Z}^{A}X^{C}$

In this notation, X takes the place of the element's symbol, A is its atomic mass, Z is its atomic number, and C is its charge (This number isn't written down if there is a zero charge). From this, we can find:

• The number of protons from looking at Z.
• The number of electrons from adding Z and C.
• The number of neutrons from subtracting Z from A.

Use this information to find the number of protons, electrons, and neutrons from the following isotopes.

1. $_{9}^{19}F$
2. $_{17}^{36}Cl^{-1}$

Section 2.11

(★) 2.49 (Area of a Rectangle) A rectangle has an area of 24 $in.^{2}$ and a length b in inches. What does the expression ${\frac {24}{b}}$ represent? What quantity does the expression $2(b+{\frac {24}{b}})$ represent?

(★) 2.50 (Change in Length) Consider an equilateral triangle with sides of length s. Find the perimeter of the triangle if we increase the lengths of the sides by 5. Find the perimeter if we double the lengths of the sides.

(★) 2.51 (Metal Wire) A metal wire of length x is bent into a square. Express the length of a side of the square in terms of x.

(★★) 2.52 (Vegetable Garden) A rectangular vegetable garden is 12 meters long and 20 meters wide. Surrounding the garden is a gravel path of width w.
1. Write an expression that can be used to find the outer perimeter of the gravel path.
2. If you measure w to be 3 meters, what is the outer perimeter of the path?

(★★) 2.53 (Racetrack) An Olympic racetrack is made up of two straight sides, each measuring 84.39 meters in length, and two semi-circular curves with a radius of 36.5 meters as pictured. The track has a width of w.
1. Write an expression that can be used to find the outer perimeter of the racetrack. (Remember that the perimeter of a circle is $2\pi r$)
2. If you measure that the width of the track is 1.22, what is the outer perimeter of the path?

(★★) 2.54 (Cube of p + q) As demonstrated in the section, the square of the expression $(p+q)$ can be represented by the diagram below.

Use a similar diagram to find the cube of the expression $(p+q)$.

(★★★) 2.55 (Cube of p + q + r)
1. Create a diagram that represents the square of $(p+q+r)$, then use it to find its value.
2. Do the same thing, but for the cube of $(p+q+r)$.

Section 2.12

(★) 2.56 (Evaluating Functions I) Evaluate $f(x)=x^{2}$ at the following values.

1. $f(2)$
2. $f(-5)$

(★) 2.57 (Evaluating Functions II) Evaluate $g(x)$ at the following values.

(★) 2.58 (Evaluating Functions III) Evaluate $h(x)$ at the following values.

(★) 2.59 (Evaluating Functions IV) Evaluate $i(x)$ at the following values.

(★) 2.60 (Evaluating Functions V) Evaluate $j(x)$ at the following values.

(★) 2.61 (Evaluating Functions VI) Evaluate $k(x)$ at the following values.

(★) 2.62 (Evaluating Functions VII) Evaluate $l(x)$ at the following values.

(★) 2.63 (Evaluating Functions VIII) Evaluate $m(x)$ at the following values.

(★) 2.64 (Functions and Relations) Which of the following definitions are functions and which are relations:

Projects

P2.1 (Pythagorean Triples)