Multiplication rule of inequalities

Multiplying the quantities on both sides of an inequality by a quantity is called the multiplication rule of inequalities.

Introduction

There are two quantities in an inequality. Multiplying a quantity on one side by a quantity and not multiplying the quantity on other side by the same quantity imbalances the inequality but multiplying the quantities on both sides of inequality by a quantity does not violate the rule of inequality. Hence, it is called the inequality multiplication rule.

$5 < 10$

$5$ and $10$ are two quantities on left and right-hand side of inequality. It expresses that the number $5$ is less than $10$.

Now, multiply the number $5$ by $4$ but do not multiply the $10$ by the number $4$.

$\implies$ $4 \times 5 < 10$

$\implies$ $20 < 10$

The mathematical statement in the above inequality is wrong because the number $20$ is not less than $10$. Hence, it should be written as follows.

$\,\,\,\therefore\,\,\,\,\,\,$ $20 > 10$

$5 < 10$ is the actual inequality but it becomes $20 > 10$ after multiplying the quantity on left-hand side of inequality by $4$. Compare the quantities in the corresponding sides of the inequalities. Nothing changed on right hand side of the inequalities but the quantities on left-hand side of inequalities are changed. Similarly, the sign between them is changed due to the improper multiplication.

It clears that multiplying the quantity on any side of the inequality by a particular quantity affects the equilibrium of the inequality. Now, multiply quantities on both sides of the inequality $5 < 10$ by $4$ to understand the multiplication rule of the inequalities.

$\implies$ $4 \times 5 < 4 \times 10$

$\,\,\,\therefore\,\,\,\,\,\,$ $20 < 40$

Now, let’s analyze the inequalities $5 < 10$ and $20 < 40$.

Unchangeability

The sign between the numbers $5$ and $10$ is less-than $<$ in the inequality. After multiplying the quantities $5$ and $10$ by $4$, the quantities become $20$ and $40$ respectively. However, the sign between them is unchanged.

Equilibrium

$5 < 10$ is the example inequality. It can be written as follows.

$\implies$ $5 < 2 \times 5$

The quantity on the right-hand side of the inequality is $10$ and it is two times greater than the quantity on the left-hand side of the inequality.

After multiplying the inequality $5 < 10$ by $4$, it becomes as $20 < 40$ and can be written as follows.

$\implies$ $20 < 2 \times 20$

The quantity on the right-hand side of the inequality is $40$ and it is also two times greater than the quantity on the left-hand side of the inequality.

Examples

The below inequalities in algebraic form expresses how to multiply the quantities on both sides of an inequality by a quantity $c$.

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

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