As we know that an equation is defined as a statement involving variable (s) and the sign of equality (=), it means left hand side is equal to right hand side.

Similarly in the inequality two quantities are unequal, but a relationship exists between them, it involves the sign of inequality and it is also called as inequation.

e.g. 2x + 3 0 (One variable)
4x + 9y > 0 (Two variable)

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Linear inequations in one variable and the solution space:
Let a be a non - zero real number and x be a variable. Then inequations of the form ax + b < 0 ,
ax + b > 0 , ax + b0 , ax + b < 0, ax + b0.

e .g. 4x + 9 0 , 3x + 10 > 30

Solution space: The set of the values of the variables that satisfy an equality or inequation are called the solution space.

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Linear Inequalities in Two Variable :
An expression of the types ax + by c or ax + by < c or ax + by c or ax + by > c are known as linear inequations or linear inequalities in two variables x and y , where a, b, and c are real numbers x and y are variables.

e.g. 4x + 6y 15,
x + y9,
2x + 3y < 21
are linear inequations in two variables x and y.

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Shading of linear inequalities

Method 1: If linear inequality has pure constant then there are two ways to shade either towards origin or away from the origin.

(a) If inequality has ≤ sign, then shade towards origin. for e.g. 4x + 6y ≤ 24

(b) If inequality has ≥ sign, then shade away from the origin. for e.g. 4x + 6y ≥24

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Method 2: Put x = 0 and y = 0 in the inequality if it satify shade towards origin otherwise away from the origin.

for e.g.

10x + 6y ≤ 60 , shade towards origin

5x + 10y ≥30, shade away from the origin.

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Note: If linear inequality has no pure constant then line will passes through origin, there are two options to shade the inequality either towards x-axis or towards y-axis.

In this case select any axis x or y, and then select any point on the selected axis put these points in linear inequality if it satisfies shade towards the selected point or vice-verse.

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