Section 1
The Real Numbers and Their Properties
Whole numbers are formed by adding the number$\_\_\_\_$ to the set of natural numbers.
The number -3 is an integer, but it is also a(n)$\_\_\_\_$ and $\mathrm{a}(\mathrm{n})$ $\_\_\_\_$ .
If $a<b$, then $a$ is to the $\_\_\_\_$ of $b$ on the number line.
If a real number is not a rational number, it is a ( n )$\_\_\_\_$ number.
True or False. If $-x$ is positive, then $-x>0$.
True or False. $\frac{-5}{2}<-2 \frac{1}{2}$
True or False. The sum of two rational numbers is a rational number.
True or False. The sum of two irrational numbers is always an irrational number.
In Exercises 9-16, Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$\frac{1}{3}$
Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$\frac{2}{3}$
Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$-\frac{4}{5}$
Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$-\frac{3}{12}$
Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$\frac{3}{11}$
Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$\frac{11}{33}$
Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$\frac{95}{30}$
Write each of the following rational numbers as a decimal and state whether the decimal is repeating or terminating.$\frac{41}{15}$
In Exercises 17-24, Convert each decimal to a quotient of two integers in lowest terms.3.75
Convert each decimal to a quotient of two integers in lowest terms. -2.35
Convert each decimal to a quotient of two integers in lowest terms.$-5 . \overline{3}$
Convert each decimal to a quotient of two integers in lowest terms.$9 . \overline{6}$
Convert each decimal to a quotient of two integers in lowest terms.$2 . \overline{13}$
Convert each decimal to a quotient of two integers in lowest terms.$3 . \overline{23}$
Convert each decimal to a quotient of two integers in lowest terms.$4.5 \overline{23}$
Convert each decimal to a quotient of two integers in lowest terms.$1.42 \overline{35}$
In Exercises 25-32, Classify each of the following numbers as rational or irrational.-207
Classify each of the following numbers as rational or irrational.-114
Classify each of the following numbers as rational or irrational.$\sqrt{81}$
Classify each of the following numbers as rational or irrational.$-\sqrt{25}$
Classify each of the following numbers as rational or irrational.$\frac{7}{2}$
Classify each of the following numbers as rational or irrational.$-\frac{15}{12}$
Classify each of the following numbers as rational or irrational.$\sqrt{12}$
Classify each of the following numbers as rational or irrational.$\sqrt{3}$
In Exercises 33-38, List all the elements of the set $$\mathrm{A}=\left\{-19,-\frac{12}{3}, 0, \sqrt{3}, 2, \sqrt{10}, \frac{17}{4}, 11\right\}$$ that arenatural numbers
List all the elements of the set $$\mathrm{A}=\left\{-19,-\frac{12}{3}, 0, \sqrt{3}, 2, \sqrt{10}, \frac{17}{4}, 11\right\}$$ that arewhole numbers
List all the elements of the set $$\mathrm{A}=\left\{-19,-\frac{12}{3}, 0, \sqrt{3}, 2, \sqrt{10}, \frac{17}{4}, 11\right\}$$ that areintegers
List all the elements of the set $$\mathrm{A}=\left\{-19,-\frac{12}{3}, 0, \sqrt{3}, 2, \sqrt{10}, \frac{17}{4}, 11\right\}$$ that arerational numbers
List all the elements of the set $$\mathrm{A}=\left\{-19,-\frac{12}{3}, 0, \sqrt{3}, 2, \sqrt{10}, \frac{17}{4}, 11\right\}$$ that areirrational numbers
List all the elements of the set $$\mathrm{A}=\left\{-19,-\frac{12}{3}, 0, \sqrt{3}, 2, \sqrt{10}, \frac{17}{4}, 11\right\}$$ that arereal numbers
In Exercises 39-50, Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$10^3$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$5^4$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression. $\left(\frac{2}{3}\right)^3$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression. $\left(\frac{5}{2}\right)^4$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$(-2)^3$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$\left(-\frac{1}{2}\right)^4$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$2 \cdot 3^4$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$5 \cdot\left(\frac{1}{3}\right)^3$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$-2 \cdot 3^4$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$-3 \cdot(-2)^4$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$-3 \cdot(-2)^5$
Each expression contains a base with an exponent greater than 1. Name the exponent and the base and evaluate the expression.$-5 \cdot(-3)^2$
In Exercises 51-60, Use inequality symbols to write the given statements symbolically. 3 is greater than -2 .
Use inequality symbols to write the given statements symbolically.-3 is less than -2 .
Use inequality symbols to write the given statements symbolically.$\frac{1}{2}$ is greater than or equal to $\frac{1}{2}$.
Use inequality symbols to write the given statements symbolically.$x$ is less than $x+1$.
Use inequality symbols to write the given statements symbolically.5 is less than or equal to $2 x$.
Use inequality symbols to write the given statements symbolically. $x-1$ is greater than 2 .
Use inequality symbols to write the given statements symbolically.$-x$ is positive.
Use inequality symbols to write the given statements symbolically. $x$ is negative.
Use inequality symbols to write the given statements symbolically.$2 x+7$ is less than or equal to 14 .
Use inequality symbols to write the given statements symbolically.$2 x+3$ is not greater than 5 .
In Exercises 61-64, Fill in the blank with one of the symbols $=,<$, or $>$ to produce a true statement.4 $\_\_\_\_$ $\frac{24}{6}$
Fill in the blank with one of the symbols $=,<$, or $>$ to produce a true statement.-5 $\_\_\_\_$ -2
Fill in the blank with one of the symbols $=,<$, or $>$ to produce a true statement.-4 $\_\_\_\_$ 0
Fill in the blank with one of the symbols $=,<$, or $>$ to produce a true statement.$-\frac{9}{2}$ $\_\_\_\_$ $-4 \frac{1}{2}$
In Exercises 65-72, Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$A \cup B$
Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$A \cap B$
Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$A \cap C$
Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$B \cup C$
Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$(B \cap C) \cup A$
Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$(A \cup C) \cap B$
Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$(A \cup B) \cap C$
Find each set if $A=\{-4,-2,0,2,4\}$, $B=\{-3,0,1,2,3,4\}$, and $C=\{-4,-3,-2,-1,0,2\}$.$(A \cup B) \cup C$
In Exercises 73-80, Find the union and the intersection of the given pairs of intervals.$I_1=(-2,3] ; I_2=(1,5)$
Find the union and the intersection of the given pairs of intervals.$I_1=[1,7] ; I_2=(3,5)$
Find the union and the intersection of the given pairs of intervals.$I_1=(-6,2) ; I_2=[2,10)$
Find the union and the intersection of the given pairs of intervals.$I_1=(-\infty,-3] ; I_2=(-3, \infty)$
Find the union and the intersection of the given pairs of intervals.$I_1=(-\infty, 5) ; I_2=[2, \infty)$
Find the union and the intersection of the given pairs of intervals.$I_1=(-2, \infty) ; I_2=(0, \infty)$
Find the union and the intersection of the given pairs of intervals.$I_1=(-\infty, 3) \cup[5, \infty) ; I_2=[-1,7]$
Find the union and the intersection of the given pairs of intervals.$I_1=(-\infty, 2) \cup(6, \infty) ; I_2=[-3,0]$
In Exercises 81-92, Rewrite each expression without absolute value bars.$|-4|$
Rewrite each expression without absolute value bars.$-1-171$
Rewrite each expression without absolute value bars.$\left|\frac{5}{-7}\right|$
Rewrite each expression without absolute value bars.$\left|\frac{-3}{5}\right|$
Rewrite each expression without absolute value bars.$|5-\sqrt{2}|$
Rewrite each expression without absolute value bars.$|\sqrt{2}-5|$
Rewrite each expression without absolute value bars.$|\sqrt{3}-2|$
Rewrite each expression without absolute value bars.$13-\pi 1$
Rewrite each expression without absolute value bars.$\frac{8}{|-8|}$
Rewrite each expression without absolute value bars.$\frac{-8}{181}$
Rewrite each expression without absolute value bars.$|5-|-7||$
Rewrite each expression without absolute value bars. ||4|-17||
In Exercises 93-100, Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=3$ and $b=8$
Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=2$ and $b=14$
Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=-6$ and $b=9$
Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=-12$ and $b=3$
Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=-20$ and $b=-6$
Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=-14$ and $b=-1$
Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=\frac{22}{7}$ and $b=-\frac{4}{7}$
Use the absolute value to express the distance between the points with coordinates $a$ and $b$ on the number line. Then determine this distance by evaluating the absolute value expression.$a=\frac{16}{5}$ and $b=-\frac{3}{5}$
In Exercises 101-108, Graph each of the given intervals on a number line and write the inequality notation for each.$(-3,1]$
Graph each of the given intervals on a number line and write the inequality notation for each.$[-6,-2)$
Graph each of the given intervals on a number line and write the inequality notation for each.$(-3, \infty)$
Graph each of the given intervals on a number line and write the inequality notation for each.$(0, \infty)$
Graph each of the given intervals on a number line and write the inequality notation for each.$(-\infty, 5]$
Graph each of the given intervals on a number line and write the inequality notation for each.$(-\infty,-1]$
Graph each of the given intervals on a number line and write the inequality notation for each.$\left(-\frac{3}{4}, \frac{9}{4}\right)$
Graph each of the given intervals on a number line and write the inequality notation for each.$\left(-3,-\frac{1}{2}\right)$
In Exercises 109-112, Use the distributive property to write each expression without parentheses.$4(x+1)$
Use the distributive property to write each expression without parentheses.$(-3)(2-x)$
Use the distributive property to write each expression without parentheses.$5(x-y+1)$
Use the distributive property to write each expression without parentheses.$2(3 x+5-y)$
In Exercises 113-116, Find the additive inverse and reciprocal of each number. (TABLE CAN'T COPY)
Find the additive inverse and reciprocal of each number.(TABLE CAN'T COPY)
In Exercises 117-128, Name the property of real numbers that justifies the given equality. All variables represent real numbers.$(-7)+7=0$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$5+(-5)=0$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$(x+2)=1-(x+2)$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$3 a=1 \cdot 3 a$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$7(x y)=(7 x) y$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$3 \cdot(6 x)=(3 \cdot 6) x$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$\frac{3}{2}\left(\frac{2}{3}\right)=1$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$2\left(\frac{1}{2}\right)=1$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$(3+x)+0=3+x$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$x(2+y)+0=x(2+y)$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$(x+5)+2 y=x+(5+2 y)$
Name the property of real numbers that justifies the given equality. All variables represent real numbers.$(3+x)+5=x+(3+5)$
In Exercises 129-150, Perform the indicated operations.$\frac{3}{5}+\frac{4}{3}$
Perform the indicated operations.$\frac{7}{10}+\frac{3}{4}$
Perform the indicated operations.$\frac{6}{5}+\frac{5}{7}$
Perform the indicated operations.$\frac{9}{2}+\frac{5}{12}$
Perform the indicated operations.$\frac{5}{6}+\frac{3}{10}$
Perform the indicated operations.$\frac{8}{15}+\frac{2}{9}$
Perform the indicated operations.$\frac{5}{8}-\frac{9}{10}$
Perform the indicated operations.$\frac{7}{8}-\frac{1}{5}$
Perform the indicated operations.$\frac{5}{9}-\frac{7}{11}$
Perform the indicated operations.$\frac{5}{8}-\frac{7}{12}$
Perform the indicated operations.$\frac{2}{5}-\frac{1}{2}$
Perform the indicated operations.$\frac{1}{4}-\frac{1}{6}$
Perform the indicated operations. $\frac{3}{4}-\frac{8}{27}$
Perform the indicated operations.$\frac{9}{7}-\frac{14}{27}$
Perform the indicated operations.$\frac{\frac{8}{5}}{\frac{16}{15}}$
Perform the indicated operations.$\frac{\frac{5}{6}}{\frac{15}{6}}$
Perform the indicated operations.$\frac{\frac{7}{8}}{\frac{21}{16}}$
Perform the indicated operations.$\frac{\frac{3}{10}}{\frac{7}{15}}$
Perform the indicated operations.$5 \cdot \frac{3}{10}-\frac{1}{2}$
Perform the indicated operations.$2 \cdot \frac{7}{2}-\frac{3}{2}$
Perform the indicated operations.$3 \cdot \frac{2}{15}-\frac{1}{3}$
Perform the indicated operations.$2 \cdot \frac{5}{3}-\frac{3}{2}$
In Exercises 151-160, Evaluate each expression for $x=3$ and $y=-5$.$2(x+y)-3 y$
Evaluate each expression for $x=3$ and $y=-5$.$-2(x+y)+5 y$
Evaluate each expression for $x=3$ and $y=-5$.$3|x|-2|y|$
Evaluate each expression for $x=3$ and $y=-5$.$7|x-y|$
Evaluate each expression for $x=3$ and $y=-5$.$\frac{x-3 y}{2}+x y$
Evaluate each expression for $x=3$ and $y=-5$.$\frac{y+3}{x}-x y$
Evaluate each expression for $x=3$ and $y=-5$.$\frac{2(1-2 x)}{y}-(-x) y$
Evaluate each expression for $x=3$ and $y=-5$.$\frac{3(2-x)}{y}-(1-x y)$
Evaluate each expression for $x=3$ and $y=-5$.$\frac{\frac{14}{x}+\frac{1}{2}}{\frac{-y}{4}}$
Evaluate each expression for $x=3$ and $y=-5$.$\frac{\frac{4}{-y}+\frac{8}{x}}{\frac{y}{2}}$
In Exercises 161-170, Correct the error in each formula.$\frac{x}{y}+\frac{x}{3}=\frac{x}{(y+3)}$
Correct the error in each formula.$(x+2)(x+3)=x+2 x+3$
Correct the error in each formula.$5(x+3)=5 x+3$
Correct the error in each formula.$(25 x)(4 x)=100 x$
Correct the error in each formula.$x-(3 y+2)=x-3 y+2$
Correct the error in each formula. $2 x-(4 y-5)=2 x-4 y-5$
Correct the error in each formula.$\frac{x+y}{x}=1+y$
Correct the error in each formula.$\frac{x+y}{x+z}=1+\frac{y}{z}$
Correct the error in each formula.$(x+1)(y+1)=x y+1$
Correct the error in each formula.$\frac{\frac{1}{x}}{\frac{1}{y}}=\frac{1}{x y}$
Let $P$ and $Q$ be two points on a number line with coordinates $a$ and $b$, respectively. Show that the point $M$ on the number line with coordinate $\frac{a+b}{2}$ is the midpoint of the line segment PQ . [Hint: Show that $d(P, M)=d(Q, M)$.]
Find 100 rational numbers between $-\frac{4}{13}$ and $\frac{7}{13}$.
Media players. Let $A=$ the set of people who own MP3 players and $B=$ the set of people who own DVD players.a. Describe the set $A \cup B$.b. Describe the set $A \cap B$.
Optional car features. The table below indicates whether certain features are offered for each of three types of 2021 cars.(TABLE CAN'T COPY)Use the roster method to describe each of the following sets.a. $A=$ cars which offer Android Autob. $B=$ cars which offer manual transmissionc. $C=$ cars which offer Apple CarPlayd. $A \cap B$e. $A \cap C$f. $A \cup B$g. $A \cup C$
Blood pressure. A group of college students had systolic blood pressure readings that ranged from a low value of 119.5 to a high value of 134.5 inclusive. Let $x$ represent the value of the systolic-blood pressure readings. Use inequalities to describe this range of values and graph the corresponding interval on a number line.
Population projections. Population projections suggest that by 2060, the number of people 65 years and older in the United States will be about 95 million. In 2020, the number of people 65 years and older in the United States was 49 million. Let $x$ represent the number (in millions) of people in the United States who are 65 years and older. Use inequalities to describe this population range from 2020 to 2060 and graph the corresponding interval on a number line.
Heart rate. For exercise to be most beneficial, the optimum heart rate for a 20-year-old person is 120 beats per minute. Use absolute value notation to write an expression that describes the difference between the heart rate achieved by each of the following 20-year-old people and the ideal exercise heart rate. Then evaluate that expression.a. Latasha: 124 beats per minuteb. Frances: 137 beats per minutec. Ignacio: 114 beats per minute
Streaming music and video. Normal-quality music streaming uses 72 MB per hour on average, and standard definition video uses about 700 MB per hour on average. Use absolute value notation to write an expression that describes the difference between this average time and the actual time Dewayne used to stream some music and video. Then evaluate that expression. (Source: https://www.androidcentral.com)a. The music used 56 MB per hour.b. The video used 380 MB per hour.
Use the formula from Example 12 for finding the temperature in degrees Celsius from the number of cricket chirps counted in 25 seconds to find the Celsius temperature if you count 42 chirps in 25 seconds.
Use the formula from Example 12 for finding the temperature in degrees Celsius from the number of cricket chirps counted in 25 seconds to find number of chirps that would be counted if the Celsius temperature is 22 degrees.
In Exercises 181 and 182, Use the fact that eating 100 grams of broccoli (a negative calorie food) actually results in a net loss of 55 calories.If a cheeseburger has 522.5 calories, how many grams of broccoli would a person have to consume to have a net gain of zero calories?
Use the fact that eating 100 grams of broccoli (a negative calorie food) actually results in a net loss of 55 calories.Maria ate 600 grams of broccoli and now wants to eat just enough French fries to get a net calorie intake of zero. If a small order of fries has 165 calories, how many orders are required?
In Exercises 183-190, State whether each statement is true or false. Justify your assertion.The opposite of an irrational number is also an irrational number.
State whether each statement is true or false. Justify your assertion.The sum of a rational number and an irrational number is an irrational number.
State whether each statement is true or false. Justify your assertion. a. The product of a rational number and an irrational number is an irrational number.b. If your answer to part(a) is "false," modify the statement to make it a true statement.
State whether each statement is true or false. Justify your assertion.The product of two irrational numbers is an irrational number-
State whether each statement is true or false. Justify your assertion. The difference of two irrational numbers is an irrational number.
State whether each statement is true or false. Justify your assertion. The quotient of two irrational numbers is an irrational number.
State whether each statement is true or false. Justify your assertion.The product of two rational numbers is a rational number.
State whether each statement is true or false. Justify your assertion.The quotient of two rational numbers is a rational number.
In Exercises 191 and 192, Use the following definition: An integer $p$ is even if $p=2 n$ for some integer $n$; an integer $q$ is odd if $q=2 k+1$, for some integer $k$. a. Show that if an integer $a$ is odd, then $a^2$ is also an odd integer.b. Show that if $b^2$ is an even integer, then $b$ is also an even integer.
Use the following definition: An integer $p$ is even if $p=2 n$ for some integer $n$; an integer $q$ is odd if $q=2 k+1$, for some integer $k$.Show that if $p^2=2 q^2$ for two integers $p$ and $q$, then $q$ is an even integer.