Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Properties of Inequality

i) Addition:

If  x, y be any real number &  c > 0 then

 \to x \leq y,  then  x+c \leq y+c

\to x \geq y, then  x+c \geq y+c

ii) Subtraction

If  x,y be any reals and  c > 0

\to x \leq y, then  x-c \leq y-c

\to x \geq y,  then  x-c \geq y-c

iii) Multiplication

(a)   x,y be any reals and  c > 0

\to x \geq y, then  x c \geq y c

\to x \leq y , then  x c \leq y c

(b)  x, y be any reals and  c < 0

 \to x \geq y, then   x c \leq y c

\to x \leq y,  then  x c \geq y c

(iv) Division: 

(a) if  x,y be any reals and  c > 0

 \to x \geq y, then  \dfrac{x}{c} \geq \dfrac {y}{c}

\to x \leq y,  then  \dfrac{x}{c} \leq \dfrac {y}{c}

(b) If  x, y be any reals and  c < 0

\to x \geq y, then \dfrac {x}{c} \leq \dfrac {y}{c}

\to x \leq y, then \dfrac{x}{c} \geq \dfrac {y}{c}

(v) Transitive:

\to  If  x \geq y  and  y \geq z , then  x \geq z

\to If  x \leq y and   y \leq z, then  x \leq z

\to If  x \geq y and  y > z, then  x > z

\to If  x \leq y and  y < z , then  x < z

(vi) Converse:

\to  If  x \leq y , then  y \geq x

\to If  x \geq y , then  y \leq x

Other Properties :

 \to  x \geq y \Leftrightarrow  -x \leq -y

\to x \leq y \Leftrightarrow -x \geq -y

\to \dfrac{x}{a} \leq \dfrac{y}{b} \implies xb \leq ya ( \text { where } a,b \neq 0)

 \to \dfrac {x}{a} \geq \dfrac {y}{b} \implies xb \geq ya( \text { where } a,b \neq 0)

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