Section 1
Sets and the Real Number Line
A _____ is a collection of items called elements.
$\mathbb{W}=\{0,1,2,3, \ldots\}$ is called the set of _____ numbers.
$\mathbb{N}=\{1,2,3, \ldots\}$ is called the set of _____ numbers.
$\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\}$ is called the set of _____.
A set can be defined using _____ - _____ notation by using a description of the set.
Listing elements in a set within set braces is called the _____ method to define a set.
Real numbers that can be expressed as a ratio of two integers are called _____ numbers.
An _____ number is a real number that cannot be expressed as a ratio of two integers.
The statement $x<y$ means that $x$ lies to the _____ of $y$ on the number line.
The _____ _____ of $x$ is denoted by $|x|$.
Write an absolute value expression to represent the distance between $a$ and $b$ on the number line: _____.
Given the expression $b^{n},$ the value of $b$ is called the _____ and $n$ is called the _____ .
The symbol $\sqrt{x}$ represents the principal _____ $\operatorname{root}$ of $x$.
The expression $\frac{0}{5}$ equals _____, whereas $\frac{5}{0}$ is _____ _____.
Write an English sentence to represent the algebraic statement.$$3 \in \mathbb{N}$$
Write an English sentence to represent the algebraic statement.$$\frac{2}{5} \in \mathbb{Q}$$
Write an English sentence to represent the algebraic statement.$$-3.1 \notin \mathbb{Z}$$
Write an English sentence to represent the algebraic statement.$$\pi \notin \mathbb{Q}$$
Write an English sentence to represent the algebraic statement.$$\mathbb{Z} \subset \mathbb{R}$$
Write an English sentence to represent the algebraic statement.$$\mathbb{Q} \subset \mathbb{R}$$
Determine whether the statement is true or false.a. $-5 \in \mathbb{N}$b. $-5 \in \mathbb{W}$c. $-5 \in \mathbb{Z}$d. $-5 \in \mathbb{Q}$
Determine whether the statement is true or false.a. $\frac{1}{3} \in \mathbb{N}$b. $\frac{1}{3} \in \mathbb{W}$c. $\frac{1}{3} \in \mathbb{Z}$d. $\frac{1}{3} \in \mathbb{Q}$
Determine whether the statement is true or false.a. $0 . \overline{25} \in \mathbb{N}$b. $0 . \overline{25} \in \mathbb{W}$c. $0 . \overline{25} \in \mathbb{Z}$d. $0 . \overline{25} \in \mathbb{Q}$
Determine whether the statement is true or false.a. $\pi \in \mathbb{Z}$b. $\pi \in \mathbb{Q}$c. $\pi \in \mathbb{H}$d. $\pi \in \mathbb{R}$
Determine whether the statement is true or false.$25 \in\{x \mid x$ is an integer and a multiple of 5$\}$
Determine whether the statement is true or false.$-7 \in\{x \mid x$ is a natural number less than 10$\}$
Determine whether the statement is true or false.$-0 . \overline{8} \in\{x \mid x$ is a rational number greater than -0.8$\}$
Determine whether the statement is true or false.$0.45 \in\{x \mid x$ is a rational number less than $0 . \overline{45}\}$
Determine whether the statement is true or false.$22 \in\{x \mid x$ is an even whole number $\}$
Determine whether the statement is true or false.$-24 \in\{x \mid x$ is an integer and a multiple of 4$\}$
Determine whether the statement is true or false.A number can be both a rational number and an irrational number.
Determine whether the statement is true or false.A number can be both an integer and a rational number.
Determine whether the statement is true or false.a. $\{-2,-4,-6\}\subset\{-6,-4,-2,0\}$b. $\{-6,-4,-2,0\}\subset\{-2,-4,-6\}$
Determine whether the statement is true or false.a. $\{\mathrm{FL}, \mathrm{GA}\} \subset\{\mathrm{FL}, \mathrm{NM}, \mathrm{GA}, \mathrm{TX}\}$b. $\{\mathrm{FL}, \mathrm{NM}, \mathrm{GA}, \mathrm{TX}\} \subset\{\mathrm{FL}, \mathrm{GA}\}$
Determine whether the statement is true or false.a. $\mathbb{Z} \subset \mathbb{W}$b. $\mathbb{W} \subset \mathbb{Z}$
Determine whether the statement is true or false.a. $\mathbb{Z} \subset \mathbb{Q}$b. $\mathbb{Q} \subset \mathbb{Z}$
Determine whether the statement is true or false.a. $\mathbb{Q} \subset \mathbb{H}$b. $\mathbb{H} \subset \mathbb{Q}$
Determine whether the statement is true or false.a. $\mathbb{H} \subset \mathbb{R}$b. $\mathbb{Q} \subset \mathbb{R}$
Refer to $A=\left\{\sqrt{5}, 0 . \overline{3}, 0.33,-0.9,-12, \frac{11}{4}, 6, \frac{\pi}{6}\right\}$Determine which elements belong to the given set. (See Example 2)a. $\mathbb{N}$b. $\mathbb{W}$c. $\mathbb{Z}$d. $\mathbb{Q}$e. $\mathbb{H}$f. $\mathbb{R}$
Refer to $B=\left\{\frac{\pi}{2}, 0,-4,0 . \overline{48}, 1,-\sqrt{13}, 9.4\right\} .$ Determinewhich elements belong to the given set.a. $\mathbb{N}$b. $\mathbb{W}$c. $\mathbb{Z}$d. $\mathbb{Q}$e. $\mathbb{H}$f. $\mathbb{R}$
Write each statement as an inequality.$a$ is at least $5 .$
Write each statement as an inequality.$b$ is at most -6.
Write each statement as an inequality.$3 c$ is no more than $9 .$
Write each statement as an inequality.$8 d$ is no less than 16.
Write each statement as an inequality.The quantity $(m+4)$ exceeds 70 .
Write each statement as an inequality.The quantity $(n-7)$ is approximately equal to 4 .
Determine whether the statement is true or false.$$3.14<\pi$$
Determine whether the statement is true or false.$$-7<-\sqrt{7}$$
Determine whether the statement is true or false.$$6.7 \geq 6.7$$
Determine whether the statement is true or false.$$-2.1 \leq-2.1$$
Determine whether the statement is true or false.$$6 . \overline{15}>6.1 \overline{5}$$
Determine whether the statement is true or false.$$2.9 \overline{3}>2 . \overline{93}$$
Determine whether the statement is true or false.$$-\frac{9}{7}<-\frac{11}{8}$$
Determine whether the statement is true or false.$$-\frac{5}{3}<-\frac{9}{5}$$
Write the interval notation and set-builder notation for each given graph. (See Example 3$)$
Graph the given set and write the corresponding interval notation. (See Example 3$)$$$\{x \mid x \leq 6\}$$
Graph the given set and write the corresponding interval notation. (See Example 3$)$$$\{x \mid x<-4\}$$
Graph the given set and write the corresponding interval notation. (See Example 3$)$$$\left\{x \mid-\frac{7}{6}<x \leq \frac{1}{3}\right\}$$
Graph the given set and write the corresponding interval notation. (See Example 3$)$$$\left\{x \mid-\frac{4}{3} \leq x<\frac{7}{4}\right\}$$
Graph the given set and write the corresponding interval notation. (See Example 3$)$$$\{x \mid 4<x\}$$
Graph the given set and write the corresponding interval notation. (See Example 3$)$$$\{x \mid-3 \leq x\}$$
Interval notation is given for several sets of real numbers. Graph the set and write the corresponding set-builder notation. (See Example 3$)$$$(-3,7]$$
Interval notation is given for several sets of real numbers. Graph the set and write the corresponding set-builder notation. (See Example 3$)$$$[-4,-1)$$
Interval notation is given for several sets of real numbers. Graph the set and write the corresponding set-builder notation. (See Example 3$)$$$(-\infty, 6.7]$$
Interval notation is given for several sets of real numbers. Graph the set and write the corresponding set-builder notation. (See Example 3$)$$$(-\infty,-3.2)$$
Interval notation is given for several sets of real numbers. Graph the set and write the corresponding set-builder notation. (See Example 3$)$$$\left[-\frac{3}{5}, \infty\right)$$
Interval notation is given for several sets of real numbers. Graph the set and write the corresponding set-builder notation. (See Example 3$)$$$\left(\frac{7}{8}, \infty\right)$$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)$$|-6|$$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)$$|-4|$$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)$$|0|$$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)$$|1|$$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)$$|\sqrt{2}-2|$$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)$$|\sqrt{6}-6|$$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)a. $|\pi-3|$b. $|3-\pi|$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)a. $|m-11|$ for $m \geq 11$b. $|m-11|$ for $m<11$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)a. $|x+2|$ for $x \geq-2$b. $|x+2|$ for $x<-2$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)a. $|t+6|$ for $t<-6$b. $|t+6|$ for $t \geq-6$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)a. $\frac{|z-5|}{z-5}$ for $z>5$b. $\frac{|z-5|}{z-5}$ for $z<5$
Simplify each expression by writing the expression without absolute value bars. (See Examples 4-5)a. $\frac{7-x}{|7-x|}$ for $x<7$b. $\frac{7-x}{|7-x|}$ for $x>7$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$1 \text { and } 6$$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$2 \text { and } 9$$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$3 \text { and }-4$$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$-8 \text { and } 2$$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$8 \text { and } \sqrt{3}$$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$11 \text { and } \sqrt{5}$$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$6 \text { and } 2 \pi$$
Write an absolute value expression to represent the distance between the two points on the number line. Then simplify without absolute value bars. (See Example 6 )$$3 \text { and } \pi$$
Simplify the expression. (See Examples 7-8)a. $4^{2}$b. $(-4)^{2}$c. $-4^{2}$d. $\sqrt{4}$e. $-\sqrt{4}$f. $\sqrt{-4}$
Simplify the expression. (See Examples 7-8)a. $9^{2}$b. $(-9)^{2}$c. $-9^{2}$d. $\sqrt{9}$e. $-\sqrt{9}$f. $\sqrt{-9}$
Simplify the expression. (See Examples 7-8)a. $\sqrt[3]{8}$b. $\sqrt[3]{-8}$c. $-\sqrt[3]{8}$d. $\sqrt{100}$e. $\sqrt{-100}$f. $-\sqrt{100}$
Simplify the expression. (See Examples 7-8)a. $\sqrt[3]{27}$b. $\sqrt[3]{-27}$c. $-\sqrt[3]{27}$d. $\sqrt{49}$e. $\sqrt{-49}$f. $-\sqrt{49}$
Simplify the expression. (See Examples 7-8)$$\left(\frac{2}{3}\right)^{3}$$
Simplify the expression. (See Examples 7-8)$$\left(\frac{4}{5}\right)^{3}$$
Simplify the expression. (See Examples 7-8)$$(-0.2)^{4}$$
Simplify the expression. (See Examples 7-8)$$(-0.1)^{4}$$
Simplify the expression. (See Examples 7-8)$$\sqrt{\frac{169}{25}}$$
Simplify the expression. (See Examples 7-8)$$\sqrt{\frac{121}{36}}$$
Simplify the expression. (See Examples 9-11)$$20-12\left(36 \div 3^{2} \div 2\right)$$
Simplify the expression. (See Examples 9-11)$$200-2^{2}\left(6 \div \frac{1}{2} \cdot 4\right)$$
Simplify the expression. (See Examples 9-11)$$6-\left\{-12+3\left[(1-6)^{2}-18\right]\right\}$$
Simplify the expression. (See Examples 9-11)$$-5-\left\{4-6\left[(2-8)^{2}-31\right]\right\}$$
Simplify the expression. (See Examples 9-11)$$\sqrt{5^{2}-3^{2}}$$
Simplify the expression. (See Examples 9-11)$$\sqrt{6^{2}+8^{2}}$$
Simplify the expression. (See Examples 9-11)$$(\sqrt{9}+\sqrt{16})^{2}$$
Simplify the expression. (See Examples 9-11)$$(\sqrt[3]{8}+\sqrt[3]{125})^{3}$$
Simplify the expression. (See Examples 9-11)$$-4 \cdot\left(\frac{2}{5}-\frac{7}{10}\right)^{2}$$
Simplify the expression. (See Examples 9-11)$$6 \cdot\left[\left(\frac{1}{3}\right)^{2}-\left(\frac{1}{2}\right)^{2}\right]$$
Simplify the expression. (See Examples 9-11)$$9-(6+|| 3-7|-8|) \div \sqrt{25}$$
Simplify the expression. (See Examples 9-11)$$8-2(4+\| 2-5|-5|) \div \sqrt{9}$$
Simplify the expression. (See Examples 9-11)$$\frac{|11-13|-4 \cdot 2}{\sqrt{12^{2}+5^{2}}-3-10}$$
Simplify the expression. (See Examples 9-11)$$\frac{(4-9)^{2}+2^{2}-3^{2}}{|-7+4|+(-12) \div 4}$$
Use the formula $W_{h}=W_{s}\left(\frac{4000}{4000+h}\right)^{2}$ to compute the weight of an object $W_{h}$ (in lb) at a height of $h$ mi above sea level. The value of $W_{s}$ is the weight of the object (in lb) at sea level.If a man weighs $200 \mathrm{lb}$ at sea level, evaluate $W_{h}=(200)\left(\frac{4000}{4000+5.5}\right)^{2}$ to determine his weight at the top of Mt. Everest. (Mt. Everest is $29,029 \mathrm{ft}$ above sea level, or approximately $5.5 \mathrm{mi}$.) Round to 1 decimal place.
Use the formula $W_{h}=W_{s}\left(\frac{4000}{4000+h}\right)^{2}$ to compute the weight of an object $W_{h}$ (in lb) at a height of $h$ mi above sea level. The value of $W_{s}$ is the weight of the object (in lb) at sea level.In $1976,$ an SR- 71 Blackbird aircraft broke the world record for altitude by an airplane (not a rocket) by reaching an altitude of $85,135 \mathrm{ft}$ (approximately $16.1 \mathrm{mi}$ ). (Source: Lockheed Martin, www.lockheedmartin.com) If the pilot weighs $175 \mathrm{lb}$ at sea level, use the formula $W_{h}=(175)\left(\frac{4000}{4000+16.1}\right)^{2}$ to determine his weight at an altitude of $16.1 \mathrm{mi} .$ Round to 1 decimal place.
Cone-shaped paper cups are used at the water cooler in many exercise facilities. Using the formula for the volume of a cone, the volume in cubic centimeters $(\mathrm{cc})$ of this cup is $V=\frac{\pi(3.8)^{2} \cdot 8}{3}$. Approximate the volume to the nearest cubic centimeter.
An inflated balloon has a volume of $6.0 \mathrm{~L}$ (liters) at sea level, where the pressure is $1.0 \mathrm{~atm}$ (atmosphere). The balloon is allowed to ascend until the pressure is 0.5 atm. During the ascent, the temperature of the gas in the balloon falls from $20^{\circ} \mathrm{C}$ to $-23^{\circ} \mathrm{C}$. Using the ideal-gas equation from chemistry, the new volume (in liters) of the gas in the balloon is $V=6.0\left(\frac{1.0}{0.5}\right)\left(\frac{250}{293}\right)$. Approximate this volume to the nearest tenth of a liter.
Explain why all terminating decimal numbers are rational numbers.
Explain why all integers are rational numbers.
When is a parenthesis used when writing interval notation?
When is a bracket used when writing interval notation?
Explain why $\mathbb{Z} \subset \mathbb{Q}$ but $\mathbb{Q} \not \subset \mathbb{Z}$.
Explain why the statement \{1\}$\subset \mathbb{Z}$ is a valid statement, but $1 \subset \mathbb{Z}$ does not make sense.
If $n>0,$ then $n-|n|=$ _____.
If $n<0,$ then $n-|n|=$_____.
If $n>0,$ then $n+|n|=$ _____ .
If $n<0,$ then $n+|n|=$ _____ .
If $n>0,$ then $-|n|=$ _____ .
If $n<0,$ then $-|n|=$ _____.
Write an inequality representing the given statement.$b$ is positive.
Write an inequality representing the given statement.$a$ is negative.
Write an inequality representing the given statement.$b$ is nonnegative.
Write an inequality representing the given statement.$a$ is not positive.
Determine the sign of the expression. Assume that $a, b,$ and $c$ are real numbers and $a<0, b>0,$ and $c<0$.$$\frac{a b^{2}}{c^{3}}$$
Determine the sign of the expression. Assume that $a, b,$ and $c$ are real numbers and $a<0, b>0,$ and $c<0$.$$\frac{a^{2} c}{b^{4}}$$
Determine the sign of the expression. Assume that $a, b,$ and $c$ are real numbers and $a<0, b>0,$ and $c<0$.$$\frac{b(a+c)^{3}}{a^{2}}$$
Determine the sign of the expression. Assume that $a, b,$ and $c$ are real numbers and $a<0, b>0,$ and $c<0$.$$\frac{(a+b)^{2}(b+c)^{4}}{b}$$
Use a calculator to approximate the expression to 2 decimal places.$$5000\left(1+\frac{0.06}{12}\right)^{(12)(5)}$$
Use a calculator to approximate the expression to 2 decimal places.$$8500\left(1+\frac{0.05}{4}\right)^{(4)(30)}$$
Use a calculator to approximate the expression to 2 decimal places.$$\frac{-3+5 \sqrt{2}}{7}$$
Use a calculator to approximate the expression to 2 decimal places.$$\frac{6-3 \sqrt{5}}{4}$$
To evaluate the expression $\frac{-3+\sqrt{3^{2}-4(-5)(2)}}{2(-5)}$ a student entered this expression on a calculator as shown. Find the error made by the student, and correct the mistake.