Precalculus

Margaret L. Lial, John Hornsby, David I. Schneider

Chapter 1

Equations and Inequalities - all with Video Answers

Educators

Section 1

Linear Equations

Problem 1

Decide whether each statement is true or false.
The solution set of $2 x+7=x-1$ is $\{-8\}$

Julie Silva

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Problem 2

Decide whether each statement is true or false.
The equation $5(x-10)=5 x-50$ is an example of an identity.

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Problem 3

Decide whether each statement is true or false.
The equations $x^{2}-4$ and $x+2-4$ are equivalent equations.

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Problem 4

Decide whether each statement is true or false.
It is possible for a linear equation to have exactly two solutions.

Carson Merrill

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Problem 5

Explain the difference between an identity and a conditional equation.

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Problem 6

Make a complete list of the steps needed to solve a linear equation. (Some equations will not require every step.)

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Problem 7

Which one is not a linear equation?
A. $5 x+7(x-1)=-3 x$
B. $9 x^{2}-4 x+3-0$
C. $7 x+8 x-13 x$
D. $04 x-.08 x-.40$

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Problem 8

In solving the equation $3(2 x-8)=6 x-24,$ a student obtains the result $0=0$ and gives the solution set $\{0\}$. Is this correct? Explain.

Sarah X

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Problem 9

Solve each equation.
$$5 x+4-3 x-4$$

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Problem 10

Solve each equation.
$$9 x+11-7 x+1$$

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Problem 11

Solve each equation.
$$6(3 x-1)=8-(10 x-14)$$

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Problem 12

Solve each equation.
$$4(-2 x+1)=6-(2 x-4)$$

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Problem 13

Solve each equation.
$$\frac{5}{6} x-2 x+\frac{4}{3}=\frac{5}{3}$$

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Problem 14

Solve each equation.
$$\frac{7}{4}+\frac{1}{5} x-\frac{3}{2}=\frac{4}{5} x$$

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Problem 15

Solve each equation.
$$3 x+5-5(x+1)-6 x+7$$

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Problem 16

Solve each equation.
$$5(x+3)+4 x-3--(2 x-4)+2$$

Sarah X

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Problem 17

Solve each equation.
$$2[x-(4+2 x)+3]-2 x+2$$

Sarah X

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Problem 18

Solve each equation.
$$4[2 x-(3-x)+5]--7 x-2$$

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Problem 19

Solve each equation.
$$\frac{1}{14}(3 x-2)=\frac{x+10}{10}$$

Sarah X

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Problem 20

Solve each equation.
$$\frac{1}{15}(2 x+5)=\frac{x+2}{9}$$

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Problem 21

Solve each equation.
$$2 x-5=1 x+7$$

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Problem 22

Solve each equation.
$$.01 x+3.1=2.03 x-2.96$$

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Problem 23

Solve each equation.
$$-4(2 x-6)+8 x-5 x+24+x$$

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Problem 24

Solve each equation.
$$-8(3 x+4)+6 x-4(x-8)+4 x$$

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Problem 25

Solve each equation.
$$.5 x+\frac{4}{3} x-x+10$$

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Problem 26

Solve each equation.
$$\frac{2}{3} x+.25 x=x+2$$

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Problem 27

Solve each equation.
$$.08 x+.06(x+12)=7.72$$

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Problem 28

Solve each equation.
$$.04(x-12)+.06 x-1.52$$

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Problem 29

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$4(2 x+7)-2 x+22+3(2 x+2)$$

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Problem 30

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$\frac{1}{2}(6 x+20)-x+4+2(x+3)$$

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Problem 31

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$2(x-8)=3 x-16$$

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Problem 32

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$-8(x+3)=-8 x-5(x+1)$$

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Problem 33

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$.3(x+2)-.5(x+2)=-.2 x-.4$$

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Problem 34

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$-.6(x-5)+.8(x-6)=.2 x-1.8$$

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Problem 35

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$4(x+7)=2(x+12)+2(x+1)$$

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Problem 36

Decide whether each equation is an identity, a conditional equation, or $a$ contradiction. Give the solution set.
$$-6(2 x+1)-3(x-4)=-15 x+1$$

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Problem 37

A student claims that the equation $5 x=4 x$ is a contradiction, since dividing both sides by $x$ leads to $5=4,$ a false statement. Explain why the student is incorrect.

Carson Merrill

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Problem 38

If $k \neq 0,$ is the equation $x+k=x$ a contradiction, a conditional equation, or an identity? Explain.

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Problem 39

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$V=l w h, \quad$ for $l \quad$ (volume of a rectangular box)

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Problem 40

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$I=P r t, \quad$ for $P \quad$ (simple interest)

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Problem 41

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$P=a+b+c,$ for $c \quad$ (perimeter of a triangle)

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Problem 42

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$P=2 l+2 w, \quad$ for $w \quad$ (perimeter of a rectangle)

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Problem 43

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$A=\frac{1}{2} h(B+b), \quad$ for $B \quad$ (arca of a trapezoid)

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Problem 44

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$A=\frac{1}{2} h(B+b), \quad$ for $h \quad$ (area of a trapezoid)

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Problem 45

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$s=2 \pi r h+2 \pi r^{2}, \quad$ for $h \quad$ (surface area of a right circular cylinder)

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Problem 46

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$s=\frac{1}{2} g t^{2}, \quad$ for $g \quad$ (distance traveled by a falling object)

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Problem 47

Solve each formula for the indicated variable. Assume that the denominator is not 0 if variables appear in the denominator.
$S=2 l w+2 w h+2 h l, \quad$ for $h \quad(\text { surface area of a rectangular box })$

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Problem 48

Refer to Exercise $45 .$ Why is it not possible to solve this formula for $r$ using the methods of this section?

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Problem 49

Solve each equation for $x$.
$$2(x-a)+b=3 x+a$$

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Problem 50

Solve each equation for $x$.
$$5 x-(2 a+c)=a(x+1)$$

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Problem 51

Solve each equation for $x$.
$$a x+b-3(x-a)$$

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Problem 52

Solve each equation for $x$.
$$4 a-a x=3 b+b x$$

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Problem 53

Solve each equation for $x$.
$$\frac{x}{a-1}=a x+3$$

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Problem 54

Solve each equation for $x$.
$$\frac{x-1}{2 a}=\frac{1}{a-b}$$

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Problem 55

Solve each equation for $x$.
$$a^{2} x+3 x=2 a^{2}$$

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Problem 56

Solve each equation for $x$.
$$a x+b^{2}=b x-a^{2}$$

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Problem 57

Solve each equation for $x$.
$$3 x=(2 x-1)(m+4)$$

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Problem 58

Solve each equation for $x$.
$$-x=(5 x+3)(3 k+1)$$

Sarah X

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Problem 59

Luis Sanchez borrowed 3150 dollars from his brother Julio to pay for books and tuition. He agreed to repay Julio in 6 months with simple annual interest at $8 \%$.
(a) How much will the interest amount to?
(b) What amount must Luis pay Julio at the end of the 6 months?

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Problem 60

Jennifer Johnston borrows 20,900 dollars from her bank to open a florist shop. She agrees to repay the money in 18 months with simple annual interest of $10.4 \%$.
(a) How much must she pay the bank in 18 months?
(b) How much of the amount in part (a) is interest?

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Problem 61

In the metric system of weights and measures, temperature is measured in degrees Celsius ('C) instead of degrees Fahrenheit ("F). To convert between the two systems, we use the equations
$$c=\frac{5}{9}(F-32) \text { and } F=\frac{9}{5} C+32$$
In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary.
$$40^{\circ} \mathrm{C}$$

Julie Silva

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Problem 62

In the metric system of weights and measures, temperature is measured in degrees Celsius ('C) instead of degrees Fahrenheit ("F). To convert between the two systems, we use the equations
$$c=\frac{5}{9}(F-32) \text { and } F=\frac{9}{5} C+32$$
In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary.
$$200^{\circ} \mathrm{C}$$

Julie Silva

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Problem 63

In the metric system of weights and measures, temperature is measured in degrees Celsius ('C) instead of degrees Fahrenheit ("F). To convert between the two systems, we use the equations
$$c=\frac{5}{9}(F-32) \text { and } F=\frac{9}{5} C+32$$
In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary.
$$59^{\circ} \mathrm{F}$$

Julie Silva

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Problem 64

In the metric system of weights and measures, temperature is measured in degrees Celsius ('C) instead of degrees Fahrenheit ("F). To convert between the two systems, we use the equations
$$c=\frac{5}{9}(F-32) \text { and } F=\frac{9}{5} C+32$$
In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary.
$$86^{\circ} \mathrm{F}$$

Julie Silva

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Problem 65

In the metric system of weights and measures, temperature is measured in degrees Celsius ('C) instead of degrees Fahrenheit ("F). To convert between the two systems, we use the equations
$$c=\frac{5}{9}(F-32) \text { and } F=\frac{9}{5} C+32$$
In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary.
$$100^{\circ} \mathrm{F}$$

Julie Silva

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Problem 66

In the metric system of weights and measures, temperature is measured in degrees Celsius ('C) instead of degrees Fahrenheit ("F). To convert between the two systems, we use the equations
$$c=\frac{5}{9}(F-32) \text { and } F=\frac{9}{5} C+32$$
In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary.
$$350^{\circ} \mathrm{F}$$

Julie Silva

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Problem 67

Work each problem.
Venus is the hottest planet with a surface temperature of $867^{\circ} \mathrm{F}$. What is this temperature in Celsius? (Source: The World Almanac, 2003 .)

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Problem 68

Work each problem.
A record low temperature of $-89.4^{\circ} \mathrm{C}$ was recorded at the Soviet Antarctica Station of Vostok on July $21,1983 .$ Find the corresponding Fahrenheit temperature. (Source: The World Almanac, 2003.)

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Problem 69

Work each problem.
The average low temperature in Montreal, Canada, for the date January 30 is $7^{\circ} \mathrm{F}$. What is the corresponding Celsius temperature to the nearest degree? (Source: www.wunderground.com)

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Problem 70

Work each problem.
The average annual temperature in Khartoum, Sudan, is approximately $26.7^{\circ} \mathrm{C}$. What is the corresponding Fahrenheit temperature to the nearest degree? (Source: wurw. world66.com/africa/sudan/climate)

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