Algebra/Chapter 2/Exercises

A set of exercises related to concepts from Chapter 2.

This set contains 56 exercises (including the Conceptual Questions)

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

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

Q2.4 (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.
Q2.5 (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?

Q2.6 (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?

Q2.7 (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"?

Q2.8 (Identifying Mathematical Statements) Which of the following sentences are statements?

Q2.9 For the following problems, state whether the given statements are the same or different.

Q2.10 (Set of Sets) Give three examples of sets whose elements are sets.

Q2.11 (Set of Sets of Sets) Give an example of a set whose elements are sets of sets.

Q2.12 (Relating Types of Numbers) Refer to the section Types of Numbers in Section 2.4. Create a Venn Diagram which shows how each of the number types listed are related to each other.

Q2.13 (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

Q2.14 (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 and what are its units of measurement? What quantity does the expression $2(b+{\frac {24}{b}})$ represent?

Exercises

Section 2.1

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

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

$a.\ x+3,\ x=3$	$b.\ 10x-3,\ x=3$	$c.\ 4a-2,\ a=9$	$d.\ {\frac {-n-1}{3}},\ n=11$	$e.\ {\frac {-b-2}{7}},\ b=5$
$f.\$	$g.\$	$h.\$	$i.\$	$j.\$
$k.\$	$l.\$	$m.\$	$n.\$	$o.\$
$p.\$	$q.\$	$r.\$	$s.\$	$t.\$

2.3 (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.

2.4 (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 diamter of a baseball.

2.5 (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.

2.6 (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.

2.7 (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.

2.8 (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.

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

2.10 (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?

2.11 (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?

2.12 (Odd Numbers) Write an expression that represents the nth odd number, O. (First odd number is 1, Second odd number is 3, and so fort1) Afterwards, use this expression to find the 143rd odd number.

2.13 (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.

2.14 (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=0 corresponds to 1999, x=1 corresponds to 2000, and so forth, and where y is the average adjusted poverty threshold. According to the model, what was the average adjusted poverty threshold in 2005? In 2012?

2.16 (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.

Section 2.2

2.17 (Writing Mathematical Sentences) Write a mathematical sentence that best represents the following.

Section 2.3

2.18 (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

2.19 (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.

2.20 (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.

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

2.21 (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.

2.22 (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.

2.23 (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.

2.24 (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.

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

2.26 (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?

Section 2.4

2.27 (Classifying Numbers) Identify the set(s) of numbers each number belongs to.

1. $\ 12$	2. $\ -7$	3. $\ 0$	4. $\ {\frac {1}{7}}$	5. $\ {\frac {16}{13}}$
6. $\ -8.08$	7. $\ -{\frac {24}{365}}$	8. $\ 0.1231231234...$	9. $\ (10)^{2}$	10. $\ 0.5{\overline {678}}$
11. $\ 2.0{\overline {505}}$	12. $\ {\frac {1}{2}}$	13. $\ {\sqrt {17}}$	14. $\ {\frac {37}{0}}$	15. $\ \pi$
16. $\ {\sqrt {9}}$	17. $\ {\sqrt {\frac {1}{36}}}$	18. $\ -5000$	19. $\ -{\sqrt {36}}$	20. $\ 5+{\sqrt {2}}$
21. $\ {\sqrt {\frac {294}{6}}}$	22. $\ 1.23456789101112...$	23. $\ {\frac {\sqrt {100-81}}{\sqrt {76}}}$	24. $\ (10.25)^{2}$	25. $\ {\sqrt {0.33}}$

2.28 (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?

2.29 (Absolute Value) Evaluate the following expressions that involve absolute values.

$a.\ |-88|$	$b.\ |3-16|$	$c.\ |-14|+|3|$	$d.\ ||-5|-3|$	$e.\ |1-{\sqrt {3}}|+|2-{\sqrt {2}}|-|{\sqrt {3}}-{\sqrt {2}}|$
$f.\ 3-2|2-10|+11$	$g.\ {\frac {|-(-5)|-|3|}{-3}}$	$h.\ {\frac {2|3*2^{2}-1|-10|-2|}{6}}$	$i.\ {\frac {(5-6)^{2}-2|3-7|}{89-3*5^{2}}}$

Section 2.6

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

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

Section 2.7

2.31 (Change in Length) Consider the triangle below, 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.

2.32 (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.

2.33 (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?

2.34 (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?

Reason and Apply

2.35 (Gauss's Trick)

1. 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. 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. $9+12+15+...+54+57+60$

2.36 (Magic Trick) Choose any number. Add 3 onto the number, then multiply the result by 2. Subtract the chosen number, then subtract 4, and then subtract the chosen number again. The number you end with is 2, isn't it? Why does this trick work?

2.37 (Exponentially Exciting) For each of the following, determine the first whole number x, greater than 1, for which the second expression is larger than the first.

$a.\ x^{3},\ 3^{x}$
$b.\ x^{4},\ 4^{x}$
$c.\ x^{5},\ 5^{x}$
$c.\ x^{6},\ 6^{x}$

2.38 ($x^{n}$ vs. $n^{x}$) On the basis of your answers to Problem 2.35, make a conjecture that appears to be true about the two expressions $x^{n}$ and $n^{x}$, where n = 3, 4, 5, 6, 7, .... and x is a whole number greater than 1.

2.39 (Difference of Squares)

$x$	$y$	$(x+y)(x-y)$	$x^{2}-y^{2}$

1. Choose two distinct values for x and y, and then fill in the first row for the table above.

2. Compare the results you got for the two expressions. What do you think the results from part a tell you about the difference of two squares?

3. Fill in the remaining rows of the table for different values of x and y, including negative numbers. Do you think your conjecture from part (b) is correct? Explain.

2.40 (Inequalities) Determine what sign values on $x$ and $y$ would make the following statements true.

$a.\ xy>0$	$b.\ x^{2}y>0$	$c.\ {\frac {x}{y}}<0$
$d.\ -{\frac {x}{y}}<0$	$e.\ -{\frac {x^{2}}{y}}>0$	$f.\ {\frac {y^{3}}{x^{2}}}>0$

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

*2.42 (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.