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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: be able to add numbers with like signs and unlike signs, understand addition with zero.

Overview

  • Addition of Numbers with Like Signs
  • Addition with Zero
  • Addition of Numbers with Unlike Signs

Addition of numbers with like signs

Let us add the two positive numbers 2 and 3. We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 3 is positive, we move 3 more units to the right.
We are now located at 5.
Thus, 2 + 3 = 5 .

A number line with arrows on each end, labeled from negative two to eight in increments of one. There is a curved arrow starting from zero, and pointing towards two. There is another curved arrow starting from two, and pointing towards five.

Summarizing, we have

( 2 positive units ) + ( 3 positive units ) = ( 5 positive units )

Now let us add the two negative numbers 2 and 3 . We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is negative, we move 2 units to the left.
Since 3 is negative, we move 3 more units to the left.
We are now located at 5 .

Thus, ( 2 ) + ( 3 ) = 5 .

A number line with arrows on each end, labeled from negative seven to three in increments of one. There is a curved arrow starting from zero, and pointing towards negative two. There is another curved arrow starting from negative two, and pointing towards negative five

Summarizing, we have

( 2 negative units ) + ( 3 negative units ) = ( 5 negative units )

These two examples suggest that

( positive number ) + ( positive number ) = ( positive number ) ( negative number ) + ( negative number ) = ( negative number )

Adding numbers with the same sign

To add two real numbers that have the same sign, add the absolute values of the numbers and associate the common sign with the sum.

Sample set a

Find the sums.

3 + 7

Add these absolute values . | 3 | = 3 | 7 | = 7 } 3 + 7 = 10 The common sign is "+ ."

3 + 7 = + 10 or 3 + 7 = 10

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( 4 ) + ( 9 )

Add these absolute values . | 4 | = 4 | 9 | = 9 } 4 + 9 = 13 The common sign is " ."

( 4 ) + ( 9 ) = 13

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Practice set a

Find the sums.

( 4 ) + ( 8 )

12

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( 36 ) + ( 9 )

45

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14 + ( 20 )

34

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2 3 + ( 5 3 )

7 3

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2.8 + ( 4.6 )

7.4

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Addition with zero

Notice that

Addition with 0

( 0 ) + ( a positive number ) = ( that same positive number ) ( 0 ) + ( a negative number ) = ( that same negative number)

The additive identity is 0

Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity .

Addition of numbers with unlike signs

Now let us perform the addition 2 + ( 6 ) . These two numbers have unlike signs. This type of addition can also be illustrated using the number line.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 6 is negative, we move, from the 2, 6 units to the left.
We are now located at 4 .

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a curved arrow starting from zero, and pointing towards two. There is another curved arrow starting from two, and pointing towards negative four.

A rule for adding two numbers that have unlike signs is suggested by noting that if the signs are disregarded, 4 can be obtained from 2 and 6 by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and 6 . Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.

Adding numbers with unlike signs

To add two real numbers that have unlike signs, subtract the smaller absolute value from the larger absolute value and associate the sign of the number with the larger absolute value with this difference.

Sample set b

Find the following sums.

7 + ( 2 )

| 7 | = 7 Larger absolute value . Sign is " + " . | 2 | = 2 Smaller absolute value .

Subtract absolute values: 7 2 = 5. Attach the proper sign: " + " .

7 + ( 2 ) = + 5 or 7 + ( 2 ) = 5

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3 + ( 11 )

| 3 | = 3 Smaller absolute value . | 11 | = 11 Larger absolute value . Sign is " " .

Subtract absolute values: 11 3 = 8. Attach the proper sign: " " .

3 + ( 11 ) = 8

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The morning temperature on a winter's day in Lake Tahoe was 12 degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?

We need to find 12 + 25 .

| 12 | = 12 Smaller absolute value . | 25 | = 25 Larger absolute value . Sign is "+" .

Subtract absolute values: 25 12 = 13. Attach the proper sign: " + " .

12 + 25 = 13

Thus, the afternoon temperature is 13 degrees.

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Use a calculator. Add 147 + 84 .                                                      Display Reads

Type 147 147 Press + / 147 Press + 147 Type 84 84 Press = 63

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Practice set b

Find the sums.

1345.6 + ( 6648.1 )

7993.7

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Exercises

Find the sums for the the following problems.

( 3 ) + ( 12 )

15

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( 4 ) + ( 8 )

12

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( 16 ) + ( 8 )

24

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( 3 ) + ( 12 )

15

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9 + ( 6 )

15

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16 + ( 9 )

25

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5 + ( 12 ) + ( 4 )

21

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1221 + ( 44 )

1265

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47.03 + ( 22.71 )

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1.998 + ( 4.086 )

6.084

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[ ( 3 ) + ( 4 ) ] + [ ( 6 ) + ( 1 ) ]

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[ ( 2 ) + ( 8 ) ] + [ ( 3 ) + ( 7 ) ]

20

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[ ( 3 ) + ( 8 ) ] + [ ( 6 ) + ( 12 ) ]

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[ ( 8 ) + ( 6 ) ] + [ ( 2 ) + ( 1 ) ]

17

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[ 4 + ( 12 ) ] + [ 12 + ( 3 ) ]

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[ 5 + ( 16 ) ] + [ 4 + ( 11 ) ]

18

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[ 2 + ( 4 ) ] + [ 17 + ( 19 ) ]

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[ 10 + ( 6 ) ] + [ 12 + ( 2 ) ]

14

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14 + [ ( 3 ) + 5 ]

16

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[ 2 + ( 7 ) ] + ( 11 )

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[ 14 + ( 8 ) ] + ( 2 )

4

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In order for a small business to break even on a project, it must have sales of $ 21 , 000 . If the amount of sales was $ 15 , 000 , how much money did this company fall short?

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Suppose a person has $ 56.00 in his checking account. He deposits $ 100.00 into his checking account by using the automatic teller machine. He then writes a check for $ 84.50 . If an error causes the deposit not to be listed into this person's account, what is this person's checking balance?

$ 28.50

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A person borrows $ 7.00 on Monday and then $ 12.00 on Tuesday. How much has this person borrowed?

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A person borrows $ 11.00 on Monday and then pays back $ 8.00 on Tuesday. How much does this person owe?

$ 3.00

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Exercises for review

( [link] ) Simplify 4 ( 7 2 6 2 3 ) 2 2 .

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( [link] ) Simplify 35 a 6 b 2 c 5 7 b 2 c 4 .

5 a 6 c

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( [link] ) Simplify ( 12 a 8 b 5 4 a 5 b 2 ) 3 .

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( [link] ) Determine the value of | 8 | .

8

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( [link] ) Determine the value of ( | 2 | + | 4 | 2 ) + | 5 | 2 .

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Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
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BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
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Sir
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Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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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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