Q55. - Algebra 2

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.

Step-by-Step Solution

Answer

The option that is true is option D.

Step 1 - 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 $a+c>b+c$.

Step 2 - 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 $a-c>b-c$.

Step 3 - 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 $ac>bc$.

Step 4 - 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, $a-c<b-c$

Step 5 - 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).