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Equality and inequality symbols

a = b a and b are equal a > b a is strictly greater than b a < b a is strictly less than b

Some variations of these symbols include

a b a is not equal to b a b a is greater than or equal to b a b a is less than or equal to b

The last five of the above symbols are inequality symbols. We can negate (change to the opposite) any of the above statements by drawing a line through the relation symbol (as in a b ), as shown below:

a is not greater than b can be expressed as either

a > b or a b.

a is not less b than can be expressed as either

a < b or a b.

a < b and a b both indicate that a is less than b .

Grouping symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in algebra are

Parentheses: ( ) Brackets: [ ] Braces: { } Bar: ¯

In a computation in which more than one operation is involved, grouping symbols help tell us which operations to perform first. If possible, we perform operations inside grouping symbols first.

Sample set b

( 4 + 17 ) 6 = 21 6 = 15

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8 ( 3 + 6 ) = 8 ( 9 ) = 72

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5 [ 8 + ( 10 4 ) ] = 5 [ 8 + 6 ] = 5 [ 14 ] = 70

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2 { 3 [ 4 ( 17 11 ) ] } = 2 { 3 [ 4 ( 6 ) ] } = 2 { 3 [ 24 ] } = 2 { 72 } = 144

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9 ( 5 + 1 ) 24 + 3 .

The fraction bar separates the two groups of numbers 9 ( 5 + 1 ) and 24 + 3 . Perform the operations in the numerator and denominator separately.

9 ( 5 + 1 ) 24 + 3 = 9 ( 6 ) 24 + 3 = 54 24 + 3 = 54 27 = 2

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Practice set b

Use the grouping symbols to help perform the following operations.

6 { 2 [ 2 ( 10 9 ) ] }

24

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The following examples show how to use algebraic notation to write each expression.

9 minus y becomes 9 y

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7 times ( x + y ) becomes 7 ( x + y )

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4 divided by 3, times z becomes ( 4 3 ) z

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( a b ) times ( b a ) divided by (2 times a ) becomes ( a b ) ( b a ) 2 a

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Introduce a variable ( any letter will do but here we’ll let x represent the number) and use appropriate algebraic symbols to write the statement: A number plus 4 is strictly greater than 6. The answer is x + 4 > 6 .

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The order of operations

Suppose we wish to find the value of 16 + 4 9 . We could

  1. add 16 and 4, then multiply this sum by 9.
    16 + 4 9 = 20 9 = 180
  2. multiply 4 and 9, then add 16 to this product.
    16 + 4 9 = 16 + 36 = 52

We now have two values for one number. To determine the correct value we must use the standard order of operations .

    Order of operations

  1. Perform all operations inside grouping symbols, beginning with the innermost set.
  2. Perform all multiplications and divisions, as you come to them, moving left-to-right.
  3. Perform all additions and subtractions, as you come to them, moving left-to-right.

As we proceed in our study of algebra, we will come upon another operation, exponentiation, that will need to be inserted before multiplication and division. (See Section [link] .)

Sample set c

USe the order of operations to find the value of each number.

16 + 4 9 Multiply first . = 16 + 36 Now add . = 52

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( 27 8 ) + 7 ( 6 + 12 ) Combine within parentheses . = 19 + 7 ( 18 ) Multiply . = 19 + 126 Now add . = 145

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8 + 2 [ 4 + 3 ( 6 1 ) ] Begin with the innermost set of grouping symbols , ( ) . = 8 + 2 [ 4 + 3 ( 5 ) ] Now work within the next set of grouping symbols , [ ] . = 8 + 2 [ 4 + 15 ] = 8 + 2 [ 19 ] = 8 + 38 = 46

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6 + 4 [ 2 + 3 ( 19 17 ) ] 18 2 [ 2 ( 3 ) + 2 ] = 6 + 4 [ 2 + 3 ( 2 ) ] 18 2 [ 6 + 2 ] = 6 + 4 [ 2 + 6 ] 18 2 [ 8 ] = 6 + 4 [ 8 ] 18 16 = 6 + 32 2 = 38 2 = 19

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Practice set c

Use the order of operations to find each value.

2 + 3 ( 18 5 2 )

26

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4 + 3 [ 2 + 3 ( 1 + 8 ÷ 4 ) ]

37

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19 + 2 { 5 + 2 [ 18 + 6 ( 4 + 1 ) ] } 5 6 3 ( 5 ) 2

17

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Exercises

For the following problems, use the order of operations to find each value.

6 ( 4 + 1 ) ÷ ( 16 ÷ 8 ) 15

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6 ( 4 1 ) + 8 ( 3 + 7 ) 20

78

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( 8 ) ( 5 ) + 2 ( 14 ) + ( 1 ) ( 10 )

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61 22 + 4 [ 3 ( 10 ) + 11 ]

203

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( 1 + 16 3 ) 7 + 5 ( 12 )

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8 ( 6 + 20 ) 8 + 3 ( 6 + 16 ) 22

29

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( 300 25 ) ÷ ( 6 3 )

91 2 3

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4 3 + 8 28 ( 3 + 17 ) + 11 ( 6 )

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2 { ( 7 + 7 ) + 6 [ 4 ( 8 + 2 ) ] }

508

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0 + 10 ( 0 ) + 15 [ 4 ( 3 ) + 1 ]

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( 4 + 7 ) ( 8 3 )

55

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( 10 + 5 ) ( 10 + 5 ) 4 ( 60 4 )

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( 5 12 1 4 ) + ( 1 6 + 2 3 )

1

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4 ( 3 5 8 15 ) + 9 ( 1 3 + 1 4 )

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0 5 + 0 1 + 0 [ 2 + 4 ( 0 ) ]

0

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0 9 + 4 0 ÷ 7 + 0 [ 2 ( 2 2 ) ]

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For the following problems, state whether the given statements are the same or different.

x y and x > y

different

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Represent the product of 3 and x five different ways.

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Represent the sum of a and b two different ways.

a + b , b + a

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For the following problems, rewrite each phrase using algebraic notation.

( x + b ) times ( x + 7 )

( x + b ) ( x + 7 )

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x divided by (7 times b )

x 7 b

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( a + b ) divided by ( a + 4 )

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For the following problems, introduce a variable (any letter will do) and use appropriate algebraic symbols towrite the given statement.

A number minus eight equals seventeen.

x 8 = 17

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Five times a number, minus one, equals zero.

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A number divided by six is greater than or equal to forty-four.

x 6 44

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Sixteen minus twice a number equals five.

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Determine whether the statements for the following problems are true or false.

6 4 ( 4 ) ( 1 ) 10

true

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5 ( 4 + 2 10 ) 110

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8 6 48 0

true

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2 [ 6 ( 1 + 4 ) 8 ] > 3 ( 11 + 6 )

false

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6 [ 4 + 8 + 3 ( 26 - 15 ) ] Is not less than or equal to 3 [ 7 ( 10 - 4 ) ]

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The number of different ways 5 people can be arranged in a row is 5 4 3 2 1 . How many ways is this?

120

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A box contains 10 computer chips. Three chips are to be chosen at random. The number of ways this can be done is

1 0 9 8 7 6 5 4 3 2 1 3 2 1 7 6 5 4 3 2 1

How many ways is this?

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The probability of obtaining four of a kind in a five-card poker hand is

13 48 ( 52 51 50 49 48 ) ÷ ( 5 4 3 2 1 )

What is this probability?

0.00024 , or 1 4165

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Three people are on an elevator in a five story building. If each person randomly selects a floor on which to get off, the probability that at least two people get off on the same floor is

1 5 4 3 5 5 5

What is this probability?

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
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50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
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how many start and codon
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what is field
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physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
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what is ogarnic chemistry
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determine the slope giving that 3y+ 2x-14=0
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Another formula for Acceleration
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a=v/t. a=f/m a
IHUMA
innocent
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pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
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how do lnternal energy measures
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Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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