Question

Which of the following statements, will not change the direction of an inequality?

• A. A positive number is multiplied to both sides of an inequality
• B. A negative number is multiplied to both sides of an inequality
• C. A positive number is divided from both sides of an inequality
• D. A negative number is divided from both sides of an inequality

Answer 1

Answer: (A), (C)

A positive number is divided from both sides of an inequality

Let's take some examples for each of the options to validate them.

$\left(A\right)6<8$
Multiply both sides of the inequality by 2
$6\times 2<8\times 2$
$\Rightarrow 12<16,$ which is true

$\therefore$ Multiplying both sides of the inequality by a positive number does not change the direction of the inequality.

$\left(B\right)6<8$
Multiply both sides of the inequality by -2
$6\times -2<8\times -2$
$\Rightarrow -12<-16,$ which is not true. To make it true, we have to change the direction of the inequality.
$\Rightarrow -12>-16$

$\therefore$ Multiplying both sides of the inequality by a negative number changes the direction of the inequality.

$\left(C\right)6<8$
Divide both sides of the inequality by 2
$\frac{6}{2}<\frac{8}{2}$
$\Rightarrow 3<4,$ which is true

$\therefore$ Dividing both sides of the inequality by a positive number does not change the direction of the inequality.

$\left(D\right)6<8$
Divide both sides of the inequality by -2
$\frac{6}{-2}<\frac{4}{-2}$
$\Rightarrow -3<-4,$ which is not true. To make it true, we have to change the direction of the inequality.
$\Rightarrow -3>-4$

$\therefore$ Dividing both sides of the inequality by a negative number changes the direction of the inequality.