Question

If $\text{a}<\text{b}$and $\text{c}<0$, state which of the following is true.

I. $\text{ac}>\text{bc}$ II. $\text{a}+\text{c}<\text{b}+\text{c}$ III. $\text{a}-\text{c}>\text{b}-\text{c}$

A.I only B. II only C. III only

D. I and II only E. I, II, and III.

Short answer

The option that is true isoption D.

Step-by-step solution

- State the addition property of inequality.

The properties of inequality are also known as axioms of inequality.

The addition property of inequality is:

For any real number $a$, $b$ and $c$.

If $a<b$, then $a+c<b+c$ and

If $a>b$, then $a+c>b+c$.

- State the subtraction property of inequality.

The subtraction property of inequality is:

For any real number $a$, $b$ and $c$.

If $a<b$, then $a-c<b-c$ and

If $a>b$, then $a-c>b-c$.

- State the multiplication property of inequality.

The multiplication property of inequality is:

For any real number $a$, b and c

Where, c is negative:

If $a<b$, then $ac>bc$and

If $a>b$, then $ac<bc$.

Where, c is positive:

If $a<b$, then $ac<bc$ and

If $a>b$, then $ac>bc$.

- State the explanation.

From the given, $a<b$and $c<0$that is c is negative.

Then using the multiplication property, $ac>bc$].

Using addition property, $a+c<b+c$and

Using subtraction property, $a-c<b-c$

- State the conclusion.

For $a<b$ and $c<0$ , the possibilities are $ac>bc$, $a+c<b+c$ and $a-c<b-c$ .

Then statement I that is $ac>bc$holds.

Statement II that is $a+c<b+c$also holds.

But Statement III that is $a-c>b-c$ does not hold.

As I and II holds, then,

Option A that is I only is not true.

Option B that is II only is not true.

Option C that is III only is not true.

Option D that is I and II only is true.

Option E that is I, II and III is not true.

Therefore, the options that is true is option D (that is I and II only).