Question

Here is my tutorial on linear inequalities

Answer 1

LINEAR INEQUATIONS :

I have seen many people getting confused in this topic : Linear inequations :

Inequations : A statement involving variable(s) and the sign of inequality viz. > , < ,\(\ge\) ,\(\le\) is called an inequality or an inequation

An equation can be linear, cubic , qudratic etc. and it may contain 1 or more than 1 variables.

for example:
\(\large{3x-2<0}\)
\(\large{2x+3\ge 0}\)
\(\large{2x^2+y+3>3}\)

Solutions of an inequation : A solution of an inequation is the value (s)of the variable (s) that makes it a true statement .

\[\large{\frac{3-2x}{5}=\frac{x}{3}-4}\]
Let us take the above equation as an example :

LHS of this inequation is \(\large{\frac{3-2x}{5}}\) and RHS is \(\large{\frac{x}{3}-4}\)
we observe that :

For x = 9 , We have LHS =\(\large{\frac{3-2*9}{5}=-3}\) and RHS =\(\large{\frac{9}{3}-4=-1}\)

Clearly : -3<-1
LHS < RHS which is true.
Hence x = 9 is one of the solution of this inequation.
Similarly we can verify that no. greater than 7 is a solution of the given inequation.

Solving a linear inequation :
Properties of inequalities

Let a, b and c be real numbers.

Transitive Property
If a < b and b < c then a < c

Addition Property
If a < b then a + c < b + c

Subtraction Property
If a < b then a - c < b - c

Multiplication Property
If a < b and c is positive then c*a < c*b
If a < b and c is negative c*a > c*b
Note:
If each inequality sign is reversed in the above properties, we obtain similar properties.
If the inequality sign < is replaced by <= ( less than or equal) or the sign > is replaced by >= ( greater than or equal ), we also obtain similar properties.

Examplex : 1
1) \(\large{2x-4\le 0}\)
\[\large{2x-4+4\le 0+4}\]
\[\large{2x\le 4}\]
\[\large{\frac{2x}{2}\le\frac{4}{2}}\]
\[\large{x\le 2}\]

2) \(\large{2x+7\ge3}\)
\[\large{2x+7-7\ge3-7}\]
\[\large{2x\le-4}\]
\[\large{\frac{2x}{2}\le\frac{-4}{2}}\]
\[\large{x\le-2}\]

some more problems regarding solving double inequality will be posted soon

Answer 2

some help is taken from net also ... properties of linear inequalities

Answer 3

@waterineyes and @nbouscal and @lgbasallote @dpaInc please have a look

Answer 4

@Calcmathlete

Answer 5

Great Job! I never knew that inequation was another name for inequality ;) If I could give you two medals, I would!

Answer 6

:) thanks a lot @Calcmathlete .

Answer 7

@naveenbabbar and @Vincent-Lyon.Fr

Answer 8

@satellite73 sir , @.Sam. @waterineyes @maheshmeghwal9

Answer 9

Very nice @mathslover
but how would we do this inequality: ?

\[\frac{2}{x}>1\]I know that this is solved like \[\frac{2}{x}-1>0 \implies \frac{2-x}{x}>0\]but why can't we do like this ?
\[\frac{2}{x}>1 \implies 2>x.\]
what is the reason behind this?

Answer 10

It is my doubt:(

Answer 11

hmn .. may be @Limitless help u :)

Answer 12

ok @waterineyes plz help:)

Answer 13

my book gave me a reason but i couldn't understand that
it said that "the second method is nt acceptable becoz we don't know that 'x' is either negative or positive."
wt is the meaning of this statement???????????????

Answer 14

@maheshmeghwal9 yes you can do that is you know that x is positive and in case of negative you can't do that..

Answer 15

if in place of is..

Answer 16

oh i see!!
gt it thanx:)

Answer 17

REMEMBER ONE THING ALWAYS:

WHEN WE MULTIPLY OF DIVIDE BOTH SIDES BY (-) OF A LINEAR INEQUATIONS,

THEN INEQUALITY SIGN REVERSES...



If x > 2

Then on multiplying (-1), we get:

-x < -2

It is because you can simply take an example:

5 > 2

Multiply (-1) both the sides,

-5 > -2 (INCORRECT)


So the correct answer is :


-5 < -2......

Answer 18

@Zarkon sir please comment or give suggestion

Answer 19

ok @waterineyes !

Answer 20

@mathslover you have done a great job...

Answer 21

thanks waterineyes :)

Answer 22

Welcome dear..

Answer 23

1st order single variable linear equations:)

Answer 24

@mathslover very good

Answer 25

thanks a lot @annas .. any more suggestions will be welcomed .. :)

Answer 26

you should write a tutorial on matrices they are very important in mathematics. but this and previous tutorial were very nice and good. nice effort by you :)

Answer 27

ok i will soon . i will try to do that .. thanks
@Zarkon sir and @apoorvk any suggestion ?

Answer 28

good job..
but one thing i'd like to see is instructions on handling inequalities involving absolute values...
too many times i've seen users loosely use the word "and" and "or" in describing the solution sets...
for example, i've seen users say the solution to |x| > 2 is "x is greater than 2 AND x is less than -2"

Answer 29

oh ! thanks a lot

Answer 30

Best Work for Os users @mathslover @mathslover

Answer 31

Thanks a lot @goformit100 please have a look to this one also
http://openstudy.com/users/mathslover#/updates/4ff461cfe4b01c7be8c7b1ea

Answer 32

also the abbreviations that i mentioned in the tutorial :

RHS - Right Hand side
LHS - Left Hand Side
for example : we have an equation
a+b=b+c
then a+b is LHS that is left hand side of the equality sign
and b+c is the RHS that is Right hand Side of the equality sign