Q1. - Algebra 2

Question

Explain why it is not necessary to state a division property for inequalities.

Step-by-Step Solution

Answer

It is not necessary to state division properties for inequalities because it is same as multiplication with a reciprocal of a number.

Step 1 - Write division properties of inequality

For any real numbers $a,b,\text{ and }c$

Let $c$ be a positive number, if $a>b$ then $\frac{a}{c}>\frac{b}{c}$ and if $a<b$ then $\frac{a}{c}<\frac{b}{c}$

Now let c be a negative number, if a>b then $\frac{a}{c}<\frac{b}{c}$ and if a<b then $\frac{a}{c}>\frac{b}{c}$

Step 2 - Write multiplication properties of inequality

For any real numbers $a,b,\text{ and }c$

Let c be a positive number, if a>b then $ac>bc$ and if a<b then ac<bc

Now let c be a negative number, if a>b then ac<bc and if a<b then $ac>bc$

Step 3 - compare both properties

From step 1 and 2,

On taking reciprocal of number c in multiplicative properties of inequalities it becomes same as division properties of inequalities.

Thus, it is not necessary to state division properties separately.