<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set b

Use the grouping symbols to help perform the following operations.

6 { 2 [ 2 ( 10 9 ) ] }

24

Got questions? Get instant answers now!

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

9 minus y becomes 9 y

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set c

Use the order of operations to find each value.

2 + 3 ( 18 5 2 )

26

Got questions? Get instant answers now!

4 + 3 [ 2 + 3 ( 1 + 8 ÷ 4 ) ]

37

Got questions? Get instant answers now!

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

17

Got questions? Get instant answers now!

Exercises

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

6 ( 4 + 1 ) ÷ ( 16 ÷ 8 ) 15

Got questions? Get instant answers now!

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

78

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

203

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

29

Got questions? Get instant answers now!

( 300 25 ) ÷ ( 6 3 )

91 2 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

508

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 4 + 7 ) ( 8 3 )

55

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

0

Got questions? Get instant answers now!

0 9 + 4 0 ÷ 7 + 0 [ 2 ( 2 2 ) ]

Got questions? Get instant answers now!

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

x y and x > y

different

Got questions? Get instant answers now!

Represent the product of 3 and x five different ways.

Got questions? Get instant answers now!

Represent the sum of a and b two different ways.

a + b , b + a

Got questions? Get instant answers now!

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

( x + b ) times ( x + 7 )

( x + b ) ( x + 7 )

Got questions? Get instant answers now!

x divided by (7 times b )

x 7 b

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Five times a number, minus one, equals zero.

Got questions? Get instant answers now!

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

x 6 44

Got questions? Get instant answers now!

Sixteen minus twice a number equals five.

Got questions? Get instant answers now!

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

6 4 ( 4 ) ( 1 ) 10

true

Got questions? Get instant answers now!

5 ( 4 + 2 10 ) 110

Got questions? Get instant answers now!

8 6 48 0

true

Got questions? Get instant answers now!

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

false

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

Questions & Answers

how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask