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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

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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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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