Question

In your own words, explain the difference between the properties of equality and the properties of inequality.

Short answer

• Multiplying or dividing equality with a negative number does not changes the equality but the inequality sign changes on multiplying or dividing with a negative number.
• Also taking reciprocals on both sides does not changes equality but the inequality sign has to be changed when taking reciprocals.

Step-by-step solution

Step 1. Explain the first difference between equality and inequality. 

Multiplying or dividing equality with a negative number does not changes the equality but the inequality sign changes on multiplying or dividing with a negative number.

Let's take an example of equality:

$2x+3=6$

Multiply both sides by$-2:$

$$\begin{gathered} -2\left(2x+3\right)=-2\times 6 \\ -4x-6=-12 \end{gathered}$$

Now, take an example of inequality:

$2x+3<6$

Multiply both sides by $-2$(reverse the inequality):

$$\begin{gathered} -2\left(2x+3\right)>-2\times 6 \\ -4x-6>-12 \end{gathered}$$

Step 2. Explain the second difference between equality and inequality. 

Taking reciprocals on both sides does not changes equality but the inequality sign has to be changed when taking reciprocals.

Let's take an example of equality:

$\frac{2x}{3}=\frac{4}{6x}$

Reciprocal:

$\frac{3}{2x}=\frac{6x}{4}$

Now, take an example of inequality:

$\frac{2x}{3}>\frac{4}{6x}$

Reciprocal:

$\frac{3}{2x}<\frac{6x}{4}$