__INEQUALITY__

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__What is Variables?__

An element, a feature, or a factor that is liable to vary or change to find any equation and represent by x, y, z etc.

- Example: 3x + 27y = 40 is variable
- Example: 2y + 5x = 12 is variable.

__What is Constant?__

A constant is special number or a real number usually or whose value is fixed in the context of use is called a constant.

- Example: 5x – 3y + 2z = 24 here is 5, 3, 2, 24 are constant value.
- Example: 4y + 7y – 9z = 53 here is 4, 7, 9, 53 are constant value.

__Example:__

6x^{2} +11x + 3 = 0

__Answer:__

__Shortcut tricks :__

This equation +6 is coefficient of x^{2}

+ 11 is coefficient of x

+3 is constant term

** Step 1:** we multiply (+6) x (+3) = +18

**we break + 18 in two parts such that addition between them is 11.**

__Step 2 :__So, +18 = 9 + 2 = 11 . and product of both factors is 18 . So, +9 and +2 = Sum of is +11

** Step 3:** Change the sign of both the factors, So +9 = -9 and +2 = -2

and divide by coefficient of x

^{2}, So we get -9 / 6 = -3 / 2 and -2 / 6 = – 1 / 3 .

__Example:__

4y^{2} + 12y + 8 = 0

__Answer:__

__Shortcut tricks:__

4 x 8 = +32

we break + 32 in two parts such that addition between them is 12.

+32 = (+8) + (+ 4) = +12 .

Change sign of both factor and divide by coefficient of y^{2} ,

So – 8 / 4 = – 2 .

– 4 / 4 = -1

- e . – 2, – 1 .

Relation between two variables

**x>y****x**__>__y**x<y****x**__<__y**x = y relation cannot be determined.**

- 5x
^{2}+ 11x + 6 = 0 - 4y
^{2}+ 10y + 6 = 0

In equation one multiply 5 and 6 get the result is 30 separate 30 as 5 and 6 which is addition of 5+6=11.

In equation one multiply 4 and 6 get the result is 24 separate 24 as 4 and 6 which is addition of 4+6=10.

and switch the sign in to negative and divide by coefficient of x^{2}. -5 / 5 = -5 and -6 / 5 = -6 / 5.

and the second equation is do same that is -4 / 4 = -1 and -6 / 4 = -3 / 2.

Now we get the solution is for x = -5 and -6 / 5.

Now we get the solution is for y = -1 and -3 / 2.

__Example:__

7x + 3y = 15……. Equation 1

10x + 5y = 10……. Equation 2

__Answer:__

At first we multiply the equation by 5 & 3

7x + 3y = 15……. Equation 1 (Multiply by 5 )

10x + 5y = 10……. Equation 2 (Multiply by 3 )

35x + 15y = 75……. Equation 1 (Multiply by 5 )

30x + 15y = 30……. Equation 2 (Multiply by 3 )

5x = 45 .

x = 9 .

We apply value of x in any equation to obtain y value

we apply in equation 1

7 x 9 + 3y = 15

63 + 3y = 15

3y = 65 – 15

y = 17 Approx

So , x = 9 and y = 17

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