We want to write a rule for solving inequalities involving multiplication and division. Let's look at an example. We know that

5

is greater than

2 .

5 > 2

Let's multiply each side of the inequality by a positive number.

Inequality	Multiply	Simplify
$5 > 2$	$5 (3) ? 2 (3)$	$15 ? 6$

Since

15

is greater than

6,

the inequality symbol should be

> .

5 > 2 ⟶ \times 3 15 > 6

Now we will multiply each side of the inequality by a negative number.

Inequality	Multiply	Simplify
$5 > 2$	$5 (- 3) ? 2 (- 3)$	$- 15 ? - 6$

Since

- 15

is less than

- 6,

we have to put there

< .

5 > 2 ⟶ \times - 3 - 15 < - 6

We see that multiplying each side of an inequality by a negative number changes the inequality symbol. Note, since dividing by a number is the same thing as multiplying by its reciprocal, a rule for solving inequalities by multiplication will also be true for inequalities involving division.

Dividing $a$ by $b$	$a \div b = \frac{a}{b}$
Multiplying $a$ by $\frac{1}{b}$	$a \cdot \frac{1}{b} = \frac{a}{b}$

Considering the points we mentioned above, we can conclude two things.

When multiplying or dividing an inequality by a positive number, we can do it as if the inequality was an equation.
When multiplying or dividing an inequality by a negative number, we have to reverse the sign.

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