Subtraction rule of inequalities

Subtracting a quantity from the quantities on both sides of an inequality is called the subtraction rule of an inequality.

Introduction

A quantity should have to be subtracted from a quantity on one side of an inequality for either simplifying or solving inequalities. Subtracting a quantity from the quantity on one side of an inequality imbalances the quantity on other side of inequality. So, a special rule is required to subtract a quantity on either side of an inequality and it is called the subtraction rule of inequalities.

The following example helps you to understand the difference rule of inequalities.

$4 < 9$

The number $4$ is less than $9$ and find their difference. $9-4 \,=\, 5$. The difference between them is $5$.

Now, subtract $3$ from the quantity on left-hand side of inequality.

$\implies$ $4-3 \,<\, 9$

$\,\,\,\therefore\,\,\,\,\,\,$ $1 \,<\, 9$

The number $1$ is less than $9$ and their difference is $9-1 \,=\, 8$.

Actually, the difference between the quantities of inequality is $5$ but now, the difference between the quantities of inequality is $8$. It clears that subtracting a quantity from the quantity on one side of inequality imbalanced the other side in inequality.

Now, subtract the number $3$ from the quantity on right hand side of inequality to understand the subtraction rule of inequality.

$\implies$ $1 \,<\, 9-3$

$\,\,\,\therefore\,\,\,\,\,\,$ $1 \,<\, 6$

The number $1$ is less than $6$ and their difference is $6-1 \,=\, 5$.

In fact, the difference between the quantities of inequality is $5$. After subtracting $3$ from both sides of quantities, the difference between quantities of inequality is also $5$. It reveals that subtracting a quantity from both sides of quantities of an inequality is a rule of subtraction.

Examples

The following inequalities in algebraic form demonstrates how to subtract a constant $c$ from the expressions in an inequality as per the subtraction 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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