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Addition rule of inequalities

Adding a quantity on one side and also adding the same quantity on other side of an inequality is called the addition rule of an inequality.

Introduction

A quantity should be added to an inequality for either simplifying or solving it in mathematics. Adding a quantity to an expression on one side of an inequality violates the state of physical balance of expressions on both sides of inequality. Hence, it is essential to add the same quantity on other side of inequality for maintaining the equilibrium. This principle is called the addition rule of an inequality.

Let’s understand the addition rule of inequalities from a simple example.

$3 > 2$

The number $3$ is greater than $2$. In this case, the difference between the two numbers is $1$.

Add $5$ to left-hand side of inequality to understand the concept of addition rule.

$\implies$ $3+5 > 2$

$\implies$ $8 > 2$

The number $8$ is greater than $2$ and the difference between them is $6$. Actually, the difference between the numbers was $1$ in our example but now, the difference between the numbers is $6$. It clears that there is no comparison between the inequalities $3 > 2$ and $8 > 2$. So, adding a quantity to an expression on one side of an inequality is improper.

Now, add $5$ to expressions of both sides of an inequality.

$\implies$ $3+5 > 2+5$

$\implies$ $8 > 7$

The number $8$ is greater than $7$. Now, the difference between the numbers is also $1$.

The inequality $3 > 2$ becomes $8 > 7$ after adding $5$ to expressions on both sides. This basic example proved that it is essential to add the same quantity to expression on other side when a quantity is added to one side of expression in an inequality. This technique is called the sum rule of inequalities.

Examples

Look at the following inequalities in algebraic form and also observe the inequalities after adding a constant $c$ as per the sum rule of inequalities.

  1. $x > y$ is an inequality, then $x+c > y+c$
  2. $x < y$ is an inequality, then $x+c > y+c$
  3. $x \ne y$ is an inequality, then $x+c \ne y+c$
  4. $x \le y$ is an inequality, then $x+c \le y+c$
  5. $x \ge y$ is an inequality, then $x+c \ge y+c$
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