Precalculus: Functions and Graphs

Earl W. Swokowski, Jeffrey A. Cole

Chapter 1

Topics from Algebra - all with Video Answers

Educators

Section 1

Real Numbers

Problem 1

If $x<0$ and $y>0,$ determine the sign of the real number.
(a) $x y$
(b) $x^{2} y$
(c) $\frac{x}{y}+x$
(d) $y-x$

Dharmendra Jain

Dharmendra Jain

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Problem 2

If $x<0$ and $y>0,$ determine the sign of the real number.
(a) $\frac{x}{y}$
(b) $x y^{2}$
(c) $\frac{x-y}{x y}$
(d) $y(y-x)$

Mukesh Devi

Numerade Educator

Problem 3

Replace the symbol \square with elther $<,>,$ or $=$ to make the resulting statement true.
(a) $-7 \square-4$
(b) $\frac{\pi}{2} \square 1.57$
(c) $\sqrt{225} \square 15$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 4

Replace the symbol \square with elther $<,>,$ or $=$ to make the resulting statement true.
(a) $-3 \square-5$
(b) $\frac{\pi}{4} \square 0.8$
(c) $\sqrt{289} \square 17$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 5

Replace the symbol \square with elther $<,>,$ or $=$ to make the resulting statement true.
(a) $\frac{1}{11} \square 0.09$
(b) $\frac{2}{3} \square 0.6666$
(c) $\frac{22}{7} \square \pi$

Dharmendra Jain

Dharmendra Jain

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Problem 6

Replace the symbol \square with elther $<,>,$ or $=$ to make the resulting statement true.
(a) $\frac{1}{7} \square 0.143$
(b) $\frac{5}{6} \square 0.833$
(c) $\sqrt{2} \square 1.4$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 7

Express the statement as an inequality.
(a) $x$ is negative.
(b) $y$ is nonnegative.
(c) $q$ is less than or equal to $\pi$.
(d) $d$ is between 4 and 2 .
(e) $t$ is not less than 5 .
(f) The negative of $z$ is not greater than 3 .
(g) The quotient of $p$ and $q$ is at most 7.
(h) The reciprocal of $w$ is at least 9 .
(i) The absolute value of $x$ is greater than 7 .

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 8

Express the statement as an inequality.
(a) $b$ is positive.
(b) $s$ is nonpositive.
(c) $w$ is greater than or equal to $-4$.
(d) $c$ is between $\frac{1}{5}$ and $\frac{1}{3}$.
(e) $p$ is not greater than $-2$.
(f) The negative of $m$ is not less than $-2$.
(g) The quotient of $r$ and $s$ is at least $\frac{1}{s}$.
(h) The reciprocal of $f$ is at most 14.
(i) The absolute value of $x$ is less than 4.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 9

Rewrite the number without using the absolute value symbol, and simplify the result.
(a) $|-3-2|$
(b) $|-5|-|2| $
(c) $|7|+|-4|$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 10

Rewrite the number without using the absolute value symbol, and simplify the result.
(a) $|-11+1|$
(b) $|6|-|-3|$
(c) $|8|+|-9|$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 11

Rewrite the number without using the absolute value symbol, and simplify the result.
(a) $(-5)|3-6|$
(b) $|-6| /(-2)$
(c) $|-7|+|4|$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 12

Rewrite the number without using the absolute value symbol, and simplify the result.
(a) ( 4)$|6-7|$
(b) $5 /|-2|$
(c) $|-1|+|-9|$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 13

Rewrite the number without using the absolute value symbol, and simplify the result.
(a) $|4-\pi|$
(b) $|\pi-4|$
$(c)|\sqrt{2}-1.5|$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 14

Rewrite the number without using the absolute value symbol, and simplify the result.
(a) $|\sqrt{3}-1.7| \quad$
(b) $|1.7-\sqrt{3}| \quad$
(c) $\left|\frac{1}{5}-\frac{1}{3}\right|$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 15

The given numbers are coordinates of points $A, B,$ and $C,$ respectively, on a coordinate line. Find the distance.
(a) $d(A, B)$
(b) $d(B, C)$
(c) $d(C, B)$
(d) $d(A, C)$
$$3,7,-5$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 16

The given numbers are coordinates of points $A, B,$ and $C,$ respectively, on a coordinate line. Find the distance.
(a) $d(A, B)$
(b) $d(B, C)$
(c) $d(C, B)$
(d) $d(A, C)$
$$-6,-2,4$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 17

The given numbers are coordinates of points $A, B,$ and $C,$ respectively, on a coordinate line. Find the distance.
(a) $d(A, B)$
(b) $d(B, C)$
(c) $d(C, B)$
(d) $d(A, C)$
$$-9,1,10$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 18

The given numbers are coordinates of points $A, B,$ and $C,$ respectively, on a coordinate line. Find the distance.
(a) $d(A, B)$
(b) $d(B, C)$
(c) $d(C, B)$
(d) $d(A, C)$
$$8,-4,-1$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 19

The two given numbers are coordinates of points $A$ and $B$, respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol.
$$x, \quad 7$$
$d(A, B)$ is less than 5

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 20

The two given numbers are coordinates of points $A$ and $B$, respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol.
$$x, \quad-\sqrt{2}$$
$d(A, B)$ is greater than 1

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 21

The two given numbers are coordinates of points $A$ and $B$, respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol.
$$x, \quad-3$$
$d(A, B)$ is at least 8

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 22

The two given numbers are coordinates of points $A$ and $B$, respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol.
$$x, \quad 4$$
$d(A, B)$ is at most 2

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 23

The two given numbers are coordinates of points $A$ and $B$, respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol.
$$4, \quad x;$$
$d(A, B)$ is not greater than 3

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 24

The two given numbers are coordinates of points $A$ and $B$, respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol.
$$-2, x$$
$d(A, B)$ is not less than 2

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 25

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$|3+x| \text { if } x<-3$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 26

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$|5-x| \text { if } x>5$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 27

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$|2-x| \text { if } x<2$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 28

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$|7+x| \text { if } x \geq-7$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 29

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$|a-b| \text { if } a<b$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 30

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$|a-b| \text { if } a>b$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 31

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$\left|x^{2}+4\right|$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 32

Rewrite the expression without using the absolute value symbol, and simplify the result.
$$\left|-x^{2}-1\right|$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 33

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$\frac{a b+a c}{a} \square b+a c$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 34

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$\frac{a b+a c}{a} \square b+c$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 35

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$\frac{b+c}{a} \square \frac{b}{a}+\frac{c}{a}$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 36

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$\frac{a+c}{b+d} \square \frac{a}{b}+\frac{c}{d}$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 37

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$(a \div b) \div c \square a \div(b \div c)$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 38

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$(a-b)-c \square a-(b-c)$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 39

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$\frac{a-b}{b-a} \square-1$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 40

Exer. 33-40: Replace the symbol $\square$ with elther = or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$-(a+b) \square-a+b$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 41

Approximate the real-number expression to four decimal places.
(a) $\left|3.2^{2}-\sqrt{3.15}\right|$
(b) $\sqrt{(15.6-1.5)^{2}+(4.3-5.4)^{2}}$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 42

Approximate the real-number expression to four decimal places.
(a) $\frac{3.42-1.29}{5.83+2.64}$
(b) $\pi^{3}$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 43

Approximate the real-number expression. Express the answer in sclentific notation accurate to four significant figures.
(a) $\frac{1.2 \times 10^{3}}{3.1 \times 10^{2}+1.52 \times 10^{3}}$
(b) $\left(1.23 \times 10^{-4}\right)+\sqrt{4.5 \times 10^{3}}$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 44

Approximate the real-number expression. Express the answer in sclentific notation accurate to four significant figures.
(a) $\sqrt{\left|3.45-1.2 \times 10^{4}\right|+10^{3}}$
(b) $\left(1.791 \times 10^{2}\right) \times\left(9.84 \times 10^{3}\right)$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 45

The point on a coordinate line corresponding to $\sqrt{2}$ may be determined by constructing a right triangle with sides of length $1,$ as shown in the figure. Determine the points that correspond to $\sqrt{3}$ and $\sqrt{5},$ respectively. (Hint: Use the Pythagorean theorem.)
CAN'T COPY THE GRAPH

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 46

A circle of radius 1 rolls along a coordinate line in the positive direction, as shown in the figure. If point $P$ is initially at the origin, find the coordinate of $P$ after one, two, and ten complete revolutions.
CAN'T COPY THE GRAPH

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 47

Geometric proofs of properties of real numbers were first given by the ancient Greeks. In order to establish the distributive property $a(b+c)=a b+a c$ for positive real numbers $a, b,$ and $c,$ find the area of the rectangle shown in the figure in two ways.
CAN'T COPY THE FIGURE

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 48

Rational approximations to square roots can be found using a formula discovered by the ancient Babylonians. Let $x_{1}$ be the first rational approximation for $\sqrt{n}$. If we let
$$x_{2}=\frac{1}{2}\left(x_{1}+\frac{n}{x_{1}}\right)$$
then $x_{2}$ will be a better approximation for $\sqrt{n},$ and we can repeat the computation with $x_{2}$ replacing $x_{1}$. Starting with $x_{1}=\frac{3}{2},$ find the next two rational approximations for $\sqrt{2}$.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 49

Express the number in scientific form.
(a) $427,000$
(b) $0.000000098$
(c) $810,000,000$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 50

Express the number in scientific form.
(a) $85,200$
(b) $0.0000055$
(c) $24,900,000$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 51

Express the number In decimal form.
(a) $8.3 \times 10^{5}$
(b) $2.9 \times 10^{-12}$
(c) $5.63 \times 10^{k}$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 52

Express the number In decimal form.
(a) $2.3 \times 10^{7}$
(b) $7.01 \times 10^{-9}$
(c) $1.23 \times 10^{10}$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 53

The mass of a hydrogen atom is approximately
$$0.0000000000000000000000017 gram.$$
Express this number in scientific form.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 54

The mass of an electron is approximately $9.1 \times 10^{-31}$ kilogram. Express this number in decimal form.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 55

In astronomy, distances to stars are measured in light years. One light year is the distance a ray of light travels in one year. If the speed of light is approximately $186,000$ miles per second, estimate the number of miles in one light year.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 56

(a) Astronomers have estimated that the Milky Way galaxy contains 100 billion stars. Express this number in scientific form.
(b) The diameter $d$ of the Milky Way galaxy is estimated as $100,000$ light years. Express $d$ in miles. (Refer to Exercise $55 .)$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 57

The number of hydrogen atoms in a mole is Avogadro's number, $6.02 \times 10^{23} .$ If one mole of the gas has a mass of 1.01 grams, estimate the mass of a hydrogen atom.

Derek Follett

Numerade Educator

Problem 58

The population dynamics of many fish are characterized by extremely high fertility rates among adults and very low survival rates among the young. A mature halibut may lay as many as 2.5 million eggs, but only $0.00035 \%$ of the offspring survive to the age of 3 years. Use scientific form to approximate the number of offspring that live to age 3

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 59

One of the longest movies ever made is a 1970 British film that runs for 48 hours. Assuming that the film speed is 24 frames per second, approximate the total number of frames in this film. Express your answer in scientific form.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 60

The number $2^{44,457}-1$ is prime. At the time that this number was determined to be prime, it took one of the world's fastest computers about 60 days to verify that it was prime. This computer was capable of performing $2 \times 10^{11}$ calculations per second. Use scientific form to estimate the number of calculations needed to perform this computation. (More recently, in $2005,2^{30,402,457}-1,$ a number containing $9,152,052$ digits, was shown to be prime.)

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 61

When a tomado passes near a building, there is a rapid drop in the outdoor pressure and the indoor pressure does not have time to change. The resulting difference is capable of causing an outward pressure of $1.4 \mathrm{Ib} / \mathrm{in}^{2}$ on the walls and ceiling of the building.
(a) Calculate the force in pounds exerted on 1 square foot of a wall.
(b) Estimate the tons of force exerted on a wall that is
8 feet high and 40 feet wide.

Dharmendra Jain

Dharmendra Jain

Numerade Educator