6.1 algebraic expressions and formulas...6 algebra: equations and inequalities > 6.1 algebraic...

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6 Algebra: Equations and Inequalities > 6.1 Algebraic Expressions and Formulas

6.1 Algebraic Expressions and Formulas

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Evaluate algebraic expressions.

2 Use mathematical models.

3 Understand the vocabulary of algebraic expressions.

4 Simplify algebraic expressions.

YOU ARE THINKING ABOUT BUYING A high-definition television. How much distance should you allow between you and the TV for pixels to be undetectable and theimage to appear smooth?

Algebraic ExpressionsLet's see what the distance between you and your TV has to do with algebra. The biggest difference between arithmetic and algebra is the use of variables in algebra. Avariable is a letter that represents a variety of different numbers. For example, we can let x represent the diagonal length, in inches, of a high-definition television. Theindustry rule for most of the current HDTVs on the market is to multiply this diagonal length by 2.5 to get the distance, in inches, at which a person with perfect vision cansee a smooth image. This can be written but it is usually expressed as Placing a number and a letter next to one another indicates multiplication.

Notice that combines the number 2.5 and the variable x using the operation of multiplication. A combination of variables and numbers using the operations ofaddition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression. Here are some examples of algebraic expressions:

Evaluating Algebraic Expressions

1 Evaluate algebraic expressions.

Evaluating an algebraic expression means finding the value of the expression for a given value of the variable. For example, we can evaluate (the idealdistance between you and your x-inch TV) for We substitute 50 for x. We obtain or 125. This means that if the diagonal length of your TV is 50inches, your distance from the screen should be 125 inches. Because this distance is or approximately 10.4 feet.

Many algebraic expressions contain more than one operation. Evaluating an algebraic expression correctly involves carefully applying the order of operations agreementthat we studied in Chapter 5.

The Order of Operations Agreement1. Perform operations within the innermost parentheses and work outward. If the algebraic expression involves a fraction, treat the numerator and the denominatoras if they were each enclosed in parentheses.

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2.5 ⋅ x, 2.5x.

2.5x

x + 2.5

▲

The variable x

increased by 2.5

x − 2.5

▲

The variable x

decreased by 2.5

2.5x

▲

2.5 times the

variable x

x

2.5

▲

The variable x

divided by 2.5

3x + 5

▲

5 more than

3 times the

variable x

+ 7.x−−√

▲

7 more than

the square root

of the variable x

2.5x

x = 50. 2.5 ⋅ 50,12 inches = 1 foot, feet,125

12

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Page 340

2. Evaluate all exponential expressions.

3. Perform multiplications and divisions as they occur, working from left to right.

4. Perform additions and subtractions as they occur, working from left to right.

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6 Algebra: Equations and Inequalities > 6.1 Algebraic Expressions and Formulas

Example 1 Evaluating an Algebraic ExpressionEvaluate for

SOLUTION

Check Point 1Evaluate for

Great Question!Is there a difference between evaluating for and evaluating for

Yes. Notice the difference between these evaluations:

• for

• for

Work carefully when evaluating algebraic expressions with exponents and negatives.

Example 2 Evaluating an Algebraic ExpressionEvaluate for

SOLUTION

We substitute for each of the two occurrences of x. Then we use the order of operations to evaluate the algebraic expression.

Check Point 2Evaluate for

Example 3 Evaluating an Algebraic ExpressionEvaluate for and



7 + 5(x − 4)3x = 6.

7 + 5(x − 4)3

= 7 + 5(6 − 4)3

= 7 + 5(2)3

= 7 + 5 (8)

= 7 + 40

= 47

Replace x with 6.

First work inside parentheses: 6 − 4 = 2.

Evaluate the exponential expression: = 2 ⋅ 2 ⋅ 2 = 8.23

Multiply: 5 (8) = 40.

Add.

8 + 6(x − 3)2x = 13.

x2 x = −6 −x2 x = 6?

x2 x = − 6

x2

= (−6)2

= (−6) (−6) = 36

−x2 x = 6

− = − = −6 ⋅ 6 = −36x2 62

▲ ▲ ▲ ▲

The negative is not inside

parentheses and is not taken

to the second power.

+ 5x − 3x2 x = − 6.

−6

=

=

=

=

=

+ 5x − 3x2

+ 5 (−6) − 3(−6)2

36 + 5 (−6) − 3

36 + (−30) − 3

6 − 3

3

This is the given algebraic expression.

Substitute − 6 for each x.

Evaluate the exponential expression:

= (−6)(−6) = 36.(−6)2

Multiply: 5 (−6) = −30.

Add and subtract from left to right.

First add: 36+ (−30) = 6.

Subtract.

+ 4x − 7x2 x = −5.

−2 + 5xy −x2 y3 x = 4 y = −2.







Page 341

SOLUTION

We substitute 4 for each x and for each y. Then we use the order of operations to evaluate the algebraic expression.

Check Point 3Evaluate for and


−2

−2 + 5xy −x2 y3

= −2 ⋅ + 5 ⋅ 4 (−2) −42 (−2)3

= −2 ⋅ 16 + 5 ⋅ 4 (−2) − (−8)

= −32 + (−40) − (−8)

= −72 − (−8)

= −64

This is the given algebraic expression.

Substitute 4 for x and − 2 for y.

Evaluate the exponential expressions:

= 4 ⋅ 4 = 16 and42

= (−2)(−2)(−2) = −8.(−2)3

Multiply: − 2 ⋅ 16 = −32 and

5(4)(−2) = 20(−2) = −40.

Add and subtract from left to right. First add:

−32+ (−40) = −72.

Subtract: − 72− (−8) = −72+ 8 = −64.

−3 + 4xy −x2 y3 x = 5 y = −1.








6 Algebra: Equations and Inequalities > 6.1 Algebraic Expressions and Formulas > Formulas and Mathematical Models

Formulas and Mathematical Models

2 Use mathematical models.

An equation is formed when an equal sign is placed between two algebraic expressions. One aim of algebra is to provide a compact, symbolic description of the world.These descriptions involve the use of formulas. A formula is an equation that uses variables to express a relationship between two or more quantities.

Here are two examples of formulas related to heart rate and exercise.

Couch-Potato Exercise Working It

The process of finding formulas to describe real-world phenomena is called mathematical modeling. Such formulas, together with the meaning assigned to thevariables, are called mathematical models. We often say that these formulas model, or describe, the relationships among the variables.

These important definitions are repeated from earlier chapters in case your course did not cover this material.

Example 4 Modeling Caloric NeedsThe bar graph in Figure 6.1 shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderatelyactive lifestyles. (Moderately active means a lifestyle that includes physical activity equivalent to walking 1.5 to 3 miles per day at 3 to 4 miles per hour, in addition tothe light physical activity associated with typical day-to-day life.)

d



Heart rate, in

beats per minute,

the difference between

220 and your age.

▼ ▼

H = (220 − a)15

▲ ▲

is of15

Heart rate, in

beats per minute,

the difference between

220 and your age.

▼ ▼

H = (220 − a)9

10

▲ ▲

is of9

10








Page 342

FIGURE 6.1 Source: USDA









Page 343

6 Algebra: Equations and Inequalities > 6.1 Algebraic Expressions and Formulas > The Vocabulary of Algebraic Expressions

The mathematical model

describes the number of calories needed per day, W, by women in age group x with moderately active lifestyles. According to the model, how many calories per dayare needed by women between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph inFigure 6.1? By how much?

SOLUTION

Because the 19–30 age range is designated as group 4, we substitute 4 for x in the given model. Then we use the order of operations to find W, the number ofcalories needed per day by women between the ages of 19 and 30.

The formula indicates that women in the 19–30 age range with moderately active lifestyles need 2078 calories per day. Figure 6.1 indicates that 2100 calories areneeded. Thus, the mathematical model underestimates caloric needs by calories, or by 22 calories per day.

Check Point 4The mathematical model

describes the number of calories needed per day, M, by men in age group x with moderately active lifestyles. According to the model, how many calories per day areneeded by men between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph in Figure 6.1?By how much?

The Vocabulary of Algebraic Expressions

3 Understand the vocabulary of algebraic expressions.

We have seen that an algebraic expression combines numbers and variables. Here is another example of an algebraic expression:

The terms of an algebraic expression are those parts that are separated by addition. For example, we can rewrite as

This expression contains three terms, namely and

The numerical part of a term is called its coefficient. In the term 7x, the 7 is the coefficient. In the term the is the coefficient.

Coefficients of 1 and are not written. Thus, the coefficient of x, meaning 1x, is 1. Similarly, the coefficient of meaning is

A term that consists of just a number is called a numerical term or a constant. The numerical term of is

The parts of each term that are multiplied are called the factors of the term. The factors of the term 7x are 7 and x.

Like terms are terms that have the same variable factors. For example, 3x and 7x are like terms.




W = −66 + 526x + 1030x2

W

W

W

W

W

= −66 + 526x + 1030x2

= −66 ⋅ + 526 ⋅ 4 + 103042

= −66 ⋅ 16 + 526 ⋅ 4 + 1030

= −1056 + 2104 + 1030

= 2078

This is the given mathematical model.

Replace each occurrence of x with 4.

Evaluate the exponential expression:

= 4 ⋅ 4 = 16.42

Multiply from left to right:

−66 ⋅ 16 = −1056 and 526 ⋅ 4 = 2104.

Add.

2100 − 2078

M = −120 + 998x + 590x2

7x − 9y − 3.

7x − 9y − 3

7x + (−9y) + (−3) .

7x, −9y, −3.

−9y, −9

−1 −y, −1y, −1.

7x − 9y − 3 −3.













6 Algebra: Equations and Inequalities > 6.1 Algebraic Expressions and Formulas > Simplifying Algebraic Expressions

Simplifying Algebraic Expressions

4 Simplify algebraic expressions.

The properties of real numbers that we discussed in Chapter 5 can be applied to algebraic expressions.

Properties of Real NumbersProperty Example

Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Addition

Associative Property of Multiplication

Distributive Property

d

Great Question!Do I have to use the distributive property to combine like terms? Can't I just do it in my head?

Yes, you can combine like terms mentally. Add or subtract the coefficients of the terms. Use this result as the coefficient of the terms' variable factor(s).

The distributive property in the form

enables us to add or subtract like terms. For example,

This process is called combining like terms.

An algebraic expression is simplified when parentheses have been removed and like terms have been combined.

Example 5 Simplifying an Algebraic ExpressionSimplify:

SOLUTION

d

Check Point 5Simplify:



a + b = b + a 13 + 7x = 7x + 13x2 x2

ab = ba x ⋅ 6 = 6x

(a + b) + c = a + (b + c) 3 + (8 + x) = (3 + 8) + x = 11 + x

(ab) c = a (bc) −2 (3x) = (−2 ⋅ 3) x = −6x

a (b + c) = ab + aca (b − c) = ab − ac

ba + ca = (b + c) a

3x + 7x

7 −y2 y2

= (3 + 7) x = 10x

= 7 − 1 = (7 − 1) = 6 .y2 y2 y2 y2

5 (3x − 7) − 6x.

7 (2x − 3) − 11x.








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Page 345



SOLUTION

d


It is not uncommon to see algebraic expressions with parentheses preceded by a negative sign or subtraction. An expression of the form can be simplified asfollows:

Do you see a fast way to obtain the simplified expression on the right? If a negative sign or a subtraction symbol appears outside parentheses, drop theparentheses and change the sign of every term within the parentheses. For example,


SOLUTION

d




6 (2 + 4x) + 10 (4 + 3x) .x2 x2

7 (4 + 3x) + 2 (5 + x) .x2 x2

− (a + b)

− (3 − 7x − 4) = −3 + 7x + 4.x2

x2

8x + 2 [5 − (x − 3)] .

6x + 4 [7 − (x − 2)] .











Blitzer BonusUsing Algebra to Measure Blood-Alcohol Concentration

The amount of alcohol in a person's blood is known as blood-alcohol concentration (BAC), measured in grams of alcohol per deciliter of blood. A BAC of 0.08,meaning 0.08%, indicates that a person has 8 parts alcohol per 10,000 parts blood. In every state in the United States, it is illegal to drive with a BAC of 0.08 or higher.

How Do I Measure My Blood-Alcohol Concentration?

Here's a formula that models BAC for a person who weighs w pounds and who has n drinks * per hour.

d

Blood-alcohol concentration can be used to quantify the meaning of “tipsy.”

BAC Effects on Behavior0.05 Feeling of well-being; mild release of inhibitions; absence of observable effects0.08 Feeling of relaxation; mild sedation; exaggeration of emotions and behavior; slight impairment of motor skills; increase in reaction time0.12 Muscle control and speech impaired; difficulty performing motor skills; uncoordinated behavior0.15 Euphoria; major impairment of physical and mental functions; irresponsible behavior; some difficulty standing, walking, and talking0.35 Surgical anesthesia; lethal dosage for a small percentage of people0.40 Lethal dosage for 50% of people; severe circulatory and respiratory depression; alcohol poisoning/overdoseSource: National Clearinghouse for Alcohol and Drug Information

Keeping in mind the meaning of “tipsy,” we can use our model to compare blood-alcohol concentrations of a 120-pound person and a 200-pound person for variousnumbers of drinks. We determined each BAC using a calculator, rounding to three decimal places.

Blood-Alcohol Concentrations of a 120-Pound Person

n (number of drinks per hour) 1 2 3 4 5 6 7 8 9 10BAC (blood-alcohol concentration) 0.029 0.059 0.088 0.117 0.145 0.174 0.202 0.230 0.258 0.286 Illegal to drive

Blood-Alcohol Concentrations of a 200-Pound Person

n (number of drinks per hour) 1 2 3 4 5 6 7 8 9 10BAC (blood-alcohol concentration) 0.018 0.035 0.053 0.070 0.087 0.104 0.121 0.138 0.155 0.171 Illegal to drive



BAC =600n

120(0.6n + 169)

BAC =600n

200(0.6n + 169)








Page 346

Like all mathematical models, the formula for BAC gives approximate rather than exact values. There are other variables that influence blood-alcohol concentrationthat are not contained in the model. These include the rate at which an individual's body processes alcohol, how quickly one drinks, sex, age, physical condition, andthe amount of food eaten prior to drinking.

* A drink can be a 12-ounce can of beer, a 5-ounce glass of wine, or a 1.5-ounce shot of liquor. Each contains approximately 14 grams, or of alcohol.


ounce,12








6 Algebra: Equations and Inequalities > 6.1 Algebraic Expressions and Formulas > Concept and Vocabulary Check

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. Finding the value of an algebraic expression for a given value of the variable is called ___________________ the expression.

2. When an equal sign is placed between two algebraic expressions, an ___________________ is formed.

3. The parts of an algebraic expression that are separated by addition are called the ___________________ of the expression.

4. In the algebraic expression 7x, 7 is called the ___________________ because it is the numerical part.

5. In the algebraic expression 7x, 7 and x are called ___________________ because they are multiplied together.

6. The algebraic expressions 3x and 7x are called ___________________ because they contain the same variable to the same power.

Exercise Set 6.1Practice ExercisesIn Exercises 1–34, evaluate the algebraic expression for the given value or values of the variables.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.



5x + 7; x = 4

9x + 6; x = 5

−7x − 5; x = −4

−6x − 13; x = −3

+ 4; x = 5x2

+ 9; x = 3x2

− 6; x = −2x2

− 11; x = −3x2

− + 4; x = 5x2

− + 9; x = 3x2

− − 6; x = −2x2

− − 11; x = −3x2

+ 4x; x = 10x2

+ 6x; x = 9x2

8 + 17; x = 5x2

7 + 25; x = 3x2

− 5x; x = −11x2

− 8x; x = −5x2

+ 5x − 6; x = 4x2

+ 7x − 4; x = 6x2

4 + 5 ; x = 9(x − 7)3

6 + 5 ; x = 8(x − 6)3

− 3(x − y) ; x = 2, y = 8x2

− 4(x − y) ; x = 3, y = 8x2

2 − 5x − 6; x = −3x2

3 − 4x − 9; x = −5x2

−5 − 4x − 11; x = −1x2

−6 − 11x − 17; x = −2x2








29.

30.

31.

32.

33.

34.

The formula

expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 35–36, use the formula to convert the given Fahrenheittemperature to its equivalent temperature on the Celsius scale.

35.

36.

A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula

describes the ball's height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 37–38.

37. What was the ball's height 2 seconds after it was kicked?

38. What was the ball's height 3 seconds after it was kicked?

In Exercises 39–40, name the property used to go from step to step each time that “(why?)” occurs.

39.

40.

In Exercises 41–62, simplify each algebraic expression.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

3 + 2xy + 5 ; x = 2, y = 3x2 y2

4 + 3xy + 2 ; x = 3, y = 2x2 y2

− − 4xy + 3 ; x = −1, y = −2x2 y3

− − 3xy + 4 ; x = −3, y = −1x2 y3

; x = −2, y = 42x+3y

x+1

; x = −2, y = 42x+y

xy−2x

C = (F − 32)5

9

50°F

86°F

h = 4+ 60t − 16t2

7 + 2(x + 9)

= 7 + (2x + 18) (why?)

= 7 + (18 + 2x) (why?)

= (7 + 18) + 2x (why?)

= 25 + 2x

= 2x + 25 (why?)

5 (x + 4) + 3x

= (5x + 20) + 3x (why?)

= (20 + 5x) + 3x (why?)

= 20 + (5x + 3x) (why?)

= 20 + (5 + 3)x (why?)

= 20 + 8x

= 8x + 20 (why?)

7x + 10x

5x + 13x

5 − 8x2 x2

7 − 10x2 x2

3 (x + 5)

4 (x + 6)

4 (2x − 3)

3 (4x − 5)

5 (3x + 4) − 4

2(5x + 4) − 3

5(3x − 2) + 12x

2 (5x − 1) + 14x

7 (3y − 5) + 2(4y + 3)

4 (2y − 6) + 3(5y + 10)

5 (3y − 2) − (7y + 2)



Page 347

56.

57.

58.

59.


4 (5y − 3) − (6y + 3)

3 (−4 + 5x)− (5x − 4 )x2 x2

2 (−5 + 3x)− (3x − 5 )x2 x2

7 − 4 [3 − (4y − 5)]









60.

61.

62.

Practice PlusIn Exercises 63–66, simplify each algebraic expression.

63.

64.

65.

66.

Application ExercisesThe maximum heart rate, in beats per minute, that you should achieve during exercise is 220 minus your age:

The bar graph shows the target heart rate ranges for four types of exercise goals. The lower and upper limits of these ranges are fractions of the maximum heart rate,220–a. Exercises 67–68 are based on the information in the graph.

d

67. If your exercise goal is to improve cardiovascular conditioning, the graph shows the following range for target heart rate, H, in beats per minute:

a. What is the lower limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?

b. What is the upper limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?

68. If your exercise goal is to improve overall health, the graph in the previous column shows the following range for target heart rate, H, in beats per minute:

a. What is the lower limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?

b. What is the upper limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?

The bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for sedentary lifestyles.(Sedentary means a lifestyle that includes only the light physical activity associated with typical day-to-day life.)



6 − 5 [8 − (2y − 4)]

8x − 3 [5 − (7 − 6x)]

7x − 4 [6 − (8 − 5x)]

18 + 4 − [6 ( − 2) + 5]x2 x2

14 + 5 − [7 ( − 2) + 4]x2 x2

2 (3 − 5) − [4 (2 − 1) + 3]x2 x2

4 (6 − 3) − [2 (5 − 1) + 1]x2 x2

220 − a.

▲

This algebraic expression gives maximum

heart rate in terms of age, a.

►H = (220 − a)Lower limit of range 7

10

►H = (220 − a).Upper limit of range 45

►H = (220 − a)Lower limit of range 12

►H = (220 − a).Upper limit of range 3

5








Page 348

dSource: USDA

Use the appropriate information displayed by the graph to solve Exercises 69–70.

69. The mathematical model

describes the number of calories needed per day, F, by females in age group x with sedentary lifestyles. According to the model, how many calories per day areneeded by females between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By howmuch?

70. The mathematical model

describes the number of calories needed per day, M, by males in age group x with sedentary lifestyles. According to the model, how many calories per day areneeded by males between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By howmuch?


F = −82 + 654x + 620x2

M = −96 + 802x + 660x2









Salary after College. In 2010, MonsterCollege surveyed 1250 U.S. college students expecting to graduate in the next several years. Respondents were asked thefollowing question:

What do you think your starting salary will be at your first job after college?

The line graph shows the percentage of college students who anticipated various starting salaries.

dSource: MonsterCollege™

The mathematical model

describes the percentage of college students, p, who anticipated a starting salary, s, in thousands of dollars. Use this information to solve Exercises 71–72.

71.

a. Use the line graph to estimate the percentage of students who anticipated a starting salary of $30 thousand.

b. Use the formula to find the percentage of students who anticipated a starting salary of $30 thousand. How does this compare with your estimate in part (a)?

72.

a. Use the line graph to estimate the percentage of students who anticipated a starting salary of $40 thousand.

b. Use the formula to find the percentage of students who anticipated a starting salary of $40 thousand. How does this compare with your estimate in part (a)?

73. Read the Blitzer Bonus on page 346. Use the formula

and replace w with your body weight. Using this formula and a calculator, compute your BAC for integers from to Round to three decimal places.According to this model, how many drinks can you consume in an hour without exceeding the legal measure of drunk driving?

Writing in Mathematics74. What is an algebraic expression? Provide an example with your description.

75. What does it mean to evaluate an algebraic expression? Provide an example with your description.

76. What is a term? Provide an example with your description.

77. What are like terms? Provide an example with your description.

78. Explain how to add like terms. Give an example.

79. What does it mean to simplify an algebraic expression?

80. An algebra student incorrectly used the distributive property and wrote If you were that student's teacher, what would you say to help thestudent avoid this kind of error?

Critical Thinking ExercisesMake Sense? In Exercises 81–84, determine whether each statement makes sense or does not make sense, and explain your reasoning.

81. I did not use the distributive property to simplify

82. The terms and both contain the variable x, so I can combine them to obtain

83. Regardless of what real numbers I substitute for x and y, I will always obtain zero when evaluating



p = −0.01 + 0.8s + 3.7s2

BAC =600n

w (0.6n + 169)

n = 1 n = 10.

3 (5x + 7) = 15x + 7.

3 (2x + 5x) .

13x2 10x 23 .x3

2 y − 2y .x2 x2









Page 349

84. A model that describes the number of lobbyists x years after 2000 cannot be used to estimate the number in 2000.

In Exercises 85–92, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

85. The term x has no coefficient.

86.

87.

88.

89.

90.

91.

92.

93. A business that manufactures small alarm clocks has weekly fixed costs of $5000. The average cost per clock for the business to manufacture x clocks isdescribed by

a. Find the average cost when and 10,000.

b. Like all other businesses, the alarm clock manufacturer must make a profit. To do this, each clock must be sold for at least more than what it costs tomanufacture. Due to competition from a larger company, the clocks can be sold for $1.50 each and no more. Our small manufacturer can only produce 2000 clocksweekly. Does this business have much of a future? Explain.


5 + 3 (x − 4) = 8 (x − 4) = 8x − 32

−x − x = −x + (−x) = 0

x − 0.02 (x + 200) = 0.98x − 4

3 + 7x = 10x

b ⋅ b = 2b

(3y − 4) − (8y − 1) = −5y − 3

−4y + 4 = −4 (y + 4)

.0.5x + 5000

x

x = 100, 1000,

50¢






6.1 algebraic expressions and formulas...6 algebra: equations and inequalities > 6.1 algebraic...

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