Question

true or false? the order of an inequality remains the same if a positive number is multiplied or divided on both sides of the inequality

Answer 1

Let's say the following inequality is multiplied by A>0:

$\begin{gathered} 7>5 \\ \\ 7A>5A\text{ True} \end{gathered}$

Notice that is the same as multiplying A by 7 and multiplying A by 5. When we multiply A by 7 the result is a number greater than when we multiply A by 5, since 7>5.

Thus, in this example, we see that the order of the inequality remains.

A different problem would be multiplying the inequality by a negative number. For example, let's multiply the original inequality by -1:

$\begin{gathered} 7>5 \\ \\ -7>-5\text{ False} \end{gathered}$

As we see, -7 is more negative than -5. Thus, -7 < -5. Then, multiplying by a negative number changes the order of the inequality.

The same happens when we divide by a positive number (the order is the same) or a negative number (the order is changed) since dividing would be the same as multiplying by a fraction.

Therefore, the statement "the order of an inequality remains the same if a positive number is multiplied or divided by on both sides of the inequality" is TRUE.