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For example, consider the inequality 3 < 7 .

For 3 < 7 , if 8 is added to both sides, we get

3 + 8 < 7 + 8. 11 < 15 True

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For 3 < 7 , if 8 is subtracted from both sides, we get

3 8 < 7 8. 5 < 1 True

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For 3 < 7 , if both sides are multiplied by 8 (a positive number), we get

8 ( 3 ) < 8 ( 7 ) 24 < 56 True

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For 3 < 7 , if both sides are multiplied by 8 (a negative number), we get

( 8 ) 3 > ( 8 ) 7

Notice the change in direction of the inequality sign.

24 > 56 True

If we had forgotten to reverse the direction of the inequality sign we would have obtained the incorrect statement 24 < 56 .

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For 3 < 7 , if both sides are divided by 8 (a positive number), we get

3 8 < 7 8 True

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For 3 < 7 , if both sides are divided by 8 (a negative number), we get

3 8 > 7 8 True ( since .375 .875 )

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Sample set b

Solve the following linear inequalities. Draw a number line and place a point at each solution.

3 x > 15 Divide both sides by 3 . The 3 is a positive number , so we need not reverse the sense of the inequality . x > 5
Thus, all numbers strictly greater than 5 are solutions to the inequality 3 x > 15 .
A number line showing all numbers strictly greater than five.

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2 y 1 16 Add 1 to both sides . 2 y 17 Divide both sides by 2. y 17 2
A number line showing all numbers less than or equal to seventeen over two.

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8 x + 5 < 14 Subtract 5 from both sides . 8 x < 9 Divide both sides by 8. We must reverse the sense of the inequality since we are dividing by a negative number . x > 9 8
A number line showing all numbers strictly greater than negative nine over eight.

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5 3 ( y + 2 ) < 6 y 10 5 3 y 6 < 6 y 10 3 y 1 < 6 y 10 9 y < 9 y > 1
A number line showing all numbers strictly greater than one.

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2 z + 7 4 6 Multiply by 4 2 z + 7 24 Notice the change in the sense of the inequality . 2 z 17 z 17 2
A number line showing all numbers less than or equal to seventeen over two.

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Practice set b

Solve the following linear inequalities.

4 x 1 15

x 4

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5 y + 16 7

y 9 5

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7 ( 4 s 3 ) < 2 s + 8

s < 29 2

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5 ( 1 4 h ) + 4 < ( 1 h ) 2 + 6

h > 1 18

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18 4 ( 2 x 3 ) 9 x

x 30

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3 b 16 4

b 64 3

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7 z + 10 12 < 1

z < 2 7

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x 2 3 5 6

x 3 2

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Compound inequalities

Compound inequality

Another type of inequality is the compound inequality . A compound inequality is of the form:

a < x < b

There are actually two statements here. The first statement is a < x . The next statement is x < b . When we read this statement we say " a is less than x ," then continue saying "and x is less than b ."

Just by looking at the inequality we can see that the number x is between the numbers a and b . The compound inequality a < x < b indicates "betweenness." Without changing the meaning, the statement a < x can be read x > a . (Surely, if the number a is less than the number x , the number x must be greater than the number a .) Thus, we can read a < x < b as " x is greater than a and at the same time is less than b ." For example:

  1. 4 < x < 9 .
    The letter x is some number strictly between 4 and 9. Hence, x is greater than 4 and, at the same time, less than 9. The numbers 4 and 9 are not included so we use open circles at these points.
    A number line showing all numbers strictly greater than four, and strictly less than nine.
  2. 2 < z < 0 .
    The z stands for some number between 2 and 0. Hence, z is greater than 2 but also less than 0.
    A number line showing all numbers strictly greater than negative two, and strictly less than zero.
  3. 1 < x + 6 < 8 .
    The expression x + 6 represents some number strictly between 1 and 8. Hence, x + 6 represents some number strictly greater than 1, but less than 8.
  4. 1 4 5 x 2 6 7 9 .
    The term 5 x 2 6 represents some number between and including 1 4 and 7 9 . Hence, 5 x 2 6 represents some number greater than or equal to 1 4 to but less than or equal to 7 9 .
    A number line showing all numbers greater than or equal to one over four, and less than or equal to seven over nine.

Consider problem 3 above, 1 < x + 6 < 8 . The statement says that the quantity x + 6 is between 1 and 8. This statement will be true for only certain values of x . For example, if x = 1 , the statement is true since 1 < 1 + 6 < 8 . However, if x = 4.9 , the statement is false since 1 < 4.9 + 6 < 8 is clearly not true. The first of the inequalities is satisfied since 1 is less than 10.9 , but the second inequality is not satisfied since 10.9 is not less than 8.

We would like to know for exactly which values of x the statement 1 < x + 6 < 8 is true. We proceed by using the properties discussed earlier in this section, but now we must apply the rules to all three parts rather than just the two parts in a regular inequality.

Sample set c

Solve 1 < x + 6 < 8 .

1 6 < x + 6 6 < 8 6 Subtract 6 from all three parts . 5 < x < 2

Thus, if x is any number strictly between 5 and 2, the statement 1 < x + 6 < 8 will be true.

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Solve 3 < 2 x 7 5 < 8 .

3 ( 5 ) < 2 x 7 5 ( 5 ) < 8 ( 5 ) Multiply each part by 5. 15 < 2 x 7 < 40 Add 7 to all three parts . 8 < 2 x < 47 Divide all three parts by 2. 4 > x > 47 2 Remember to reverse the direction of the inequality signs . 47 2 < x < 4 It is customary (but not necessary) to write the inequality so that inequality arrows point to the left .

Thus, if x is any number between 47 2 and 4, the original inequality will be satisfied.

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Practice set c

Find the values of x that satisfy the given continued inequality.

4 < x 5 < 12

9 < x < 17

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3 < 7 y + 1 < 18

4 7 < y < 17 7

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0 1 6 x 7

1 x 1 6

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5 2 x + 1 3 10

8 x 29 2

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9 < 4 x + 5 2 < 14

23 4 < x < 33 4

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Does 4 < x < 1 have a solution?

no

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Exercises

For the following problems, solve the inequalities.

y + 19 2

y 17

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5 x 20

x 4

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12 b 5 < 24

b > 10

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8 x 5 > 6

x < 15 4

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21 y 8 < 2

y > 16 21

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7 y + 10 4

y 2

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3 x 15 30

x 15

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2 y + 4 3 2 3

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5 ( 2 x 5 ) 15

x 4

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6 ( 3 x 7 ) 48

x 5

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3 ( x + 3 ) > 27

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4 ( y + 3 ) > 0

y < 3

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7 ( x 77 ) 0

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3 x + 2 2 x 5

x 7

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3 x 12 7 x + 4

x 4

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x 4 > 3 x + 12

x > 8

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5 y 14

y 9

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3 [ 4 + 5 ( x + 1 ) ] < 3

x < 2

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2 [ 6 + 2 ( 3 x 7 ) ] 4

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7 [ 3 4 ( x 1 ) ] 91

x 3

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2 ( 4 x 1 ) < 3 ( 5 x + 8 )

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5 ( 3 x 2 ) > 3 ( x 15 ) + 1

x < 2

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Use a calculator to solve this equation. .0091 x 2.885 x 12.014

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What numbers satisfy the condition: twice a number plus one is greater than negative three?

x > 2

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What numbers satisfy the condition: eight more than three times a number is less than or equal to fourteen?

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One number is five times larger than another number. The difference between these two numbers is less than twenty-four. What are the largest possible values for the two numbers? Is there a smallest possible value for either number?

First number: any number strictly smaller that 6.
Second number: any number strictly smaller than 30.
No smallest possible value for either number.
No largest possible value for either number.

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The area of a rectangle is found by multiplying the length of the rectangle by the width of the rectangle. If the length of a rectangle is 8 feet, what is the largest possible measure for the width if it must be an integer (positive whole number) and the area must be less than 48 square feet?

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Exercises for review

( [link] ) Simplify ( x 2 y 3 z 2 ) 5 .

x 10 y 15 z 10

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( [link] ) Simplify [ ( | 8 | ) ] .

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( [link] ) Find the product. ( 2 x 7 ) ( x + 4 ) .

2 x 2 + x 28

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( [link] ) Twenty-five percent of a number is 12.32 . What is the number?

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( [link] ) The perimeter of a triangle is 40 inches. If the length of each of the two legs is exactly twice the length of the base, how long is each leg?

16 inches

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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