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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: understand the geometric and algebraic definitions of absolute value.

Overview

  • Geometric Definition of Absolute Value
  • Algebraic Definition of Absolute Value

Geometric definition of absolute value

Absolute value—geometric approach

The absolute value of a number a , denoted | a | , is the distance from a to 0 on the number line.

Absolute value speaks to the question of "how far," and not "which way." The phrase how far implies length, and length is always a nonnegative (zero or positive) quantity. Thus, the absolute value of a number is a nonnegative number. This is shown in the following examples:

| 5 | = 5 .
The quantity on the left side of the equal sign is read as "negative the absolute value of 5." The absolute value of 5 is 5. Hence, negative the absolute value of 5 is 5 .

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| 3 | = 3 .
The quantity on the left side of the equal sign is read as "negative the absolute value of 3 ." The absolute value of 3 is 3. Hence, negative the absolute value of 3 is ( 3 ) = 3 .

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Algebraic definition of absolute value

The problems in the first example may help to suggest the following algebraic definition of absolute value. The definition is interpreted below. Examples follow the interpretation.

Absolute value—algebraic approach

The absolute value of a number a is

| a | = { a if a 0 a if a < 0

The algebraic definition takes into account the fact that the number a could be either positive or zero ( 0 ) or negative ( < 0 ) .

  1. If the number a is positive or zero ( 0 ) , the first part of the definition applies. The first part of the definition tells us that if the number enclosed in the absolute bars is a nonnegative number, the absolute value of the number is the number itself.
  2. If the number a is negative ( < 0 ) , the second part of the definition applies. The second part of the definition tells us that if the number enclosed within the absolute value bars is a negative number, the absolute value of the number is the opposite of the number. The opposite of a negative number is a positive number.

Sample set a

Use the algebraic definition of absolute value to find the following values.

| 8 | .
The number enclosed within the absolute value bars is a nonnegative number so the first part of the definition applies. This part says that the absolute value of 8 is 8 itself.

| 8 | = 8

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| 3 | .
The number enclosed within absolute value bars is a negative number so the second part of the definition applies. This part says that the absolute value of 3 is the opposite of 3 , which is ( 3 ) . By the double-negative property, ( 3 ) = 3 .

| 3 | = 3

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Practice set a

Use the algebraic definition of absolute value to find the following values.

Exercises

For the following problems, determine each of the values.

| ( 8 ) |

8

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| ( 1 ) |

1

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| ( 17 | 12 | ) |

5

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| ( 46 | 24 | ) |

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( | 6 | + | 4 | ) 2

100

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( | 4 | + | 6 | ) 2 ( | 2 | ) 3

92

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[ | 10 | 6 ] 2

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[ ( | 4 | + | 3 | ) 3 ] 2

1

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A Mission Control Officer at Cape Canaveral makes the statement "lift-off, T minus 50 seconds." How long before lift-off?

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Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $ 2 , 400 , 000 . Use absolute value notation to describe this company's debt.

| $ 2 , 400 , 000 |

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A particular machine is set correctly if upon action its meter reads 0 units. One particular machine has a meter reading of 1.6 upon action. How far is this machine off its correct setting?

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Exercises for review

( [link] ) Write the following phrase using algebraic notation: "four times ( a + b ) ."

4 ( a + b )

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( [link] ) Is there a smallest natural number? If so, what is it?

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( [link] ) Name the property of real numbers that makes 5 + a = a + 5 a true statement.

commutative property of addition

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( [link] ) Find the quotient of x 6 y 8 x 4 y 3 .

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( [link] ) Simplify ( 4 ) .

4

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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