Chapter A.9, Problem 131AYU

To determine

To find: How would you explain to a fellow student the underlying reason for the multiplication properties for inequalities (page A78)? That is, how would you explain that the sense, or direction, of an inequality remains the same if each side is multiplied by a positive real number, whereas the direction is reversed if each side is multiplied by a negative real number?

Answer to Problem 131AYU

Multiplying by a Positive Number

The sense of the inequality is not changed if both sides are multiplied or divided by the same positive number.

Example:

Using the inequality:

$8 < 15$

Multiplying both sides by 2 gives

$8 \times 2 < 15 \times 2$

$16 < 30$ which is still true

Multiplying by a Negative Number

The sense of the inequality is reversed if both sides are multiplied or divided by the same negative number.

Example

We start with the inequality $4 >-2$.

Multiplying both sides by $-3$ gives

$4 \times -3 > -2 \times -3$

$-12 > 6$ which is not true

Hence the correct solution should be

$\begin{array}{l}4 > -2\\ 4 \times -3 < -2 \times -3\end{array}$

$-12 < 6$ (Note the change in the sign used)

Explanation of Solution

Multiplying by a Positive Number

The sense of the inequality is not changed if both sides are multiplied or divided by the same positive number.

Example:

Using the inequality:

$8 < 15$

Multiplying both sides by 2 gives

$8 \times 2 < 15 \times 2$

$16 < 30$ which is still true

Multiplying by a Negative Number

The sense of the inequality is reversed if both sides are multiplied or divided by the same negative number.

Example

We start with the inequality $4 > -2$.

Multiplying both sides by $-3$ gives

$4 \times -3 > -2 \times -3$

$-12 > 6$ which is not true

Hence the correct solution should be

$\begin{array}{l}4 > -2\\ 4 \times -3 < -2 \times -3\end{array}$

$-12 < 6$ (Note the change in the sign used)