1. ## Positive a/b

If the quotient a/b is positive, which of the following must be true?

A. a > 0
B. b > 0
C. ab > 0
D. a - b > 0
E. a + b > 0

My Work:

If the given quotient is positive, then a/b > 0.
So, the following is possible:

a/b > 0

b(a/b) > b•0

Conclusion: a > 0 is my answer.
The book's answer is C. Why?

2. ## Re: Positive a/b

Originally Posted by harpazo
If the quotient a/b is positive, which of the following must be true?

A. a > 0
B. b > 0
C. ab > 0
D. a - b > 0
E. a + b > 0

My Work:

If the given quotient is positive, then a/b > 0.
So, the following is possible:

a/b > 0

b(a/b) > b•0

Conclusion: a > 0 is my answer.
The book's answer is C. Why?
If $a=-4~\&~b=-3$ it is true that $\dfrac{-4}{-3}>0$ now you check out each in that list.

3. ## Re: Positive a/b

Please recall that multiplying both sides of an inequality, like $\displaystyle \frac{a}{b}> 1$, by a retains the direction of the inequality only if a is positive. If a is negative, the inequality is reversed.

Given that $\displaystyle \frac{a}{b}> 0$ and a is positive then $\displaystyle \left(\frac{b}{a}\right)a= b> 0(a)= 0$ so b is also positive. But if a is negative then $\displaystyle \left(\frac{a}{b}\right)a= b< 0(a)= 0$. That is, if $\displaystyle \frac{b}{a}> 0$ then either a and b are both positive or a and b are both negative.

4. ## Re: Positive a/b

Originally Posted by HallsofIvy
Please recall that multiplying both sides of an inequality, like [tex]\frac{a}{b}> 1[tex], by a retains the direction of the inequality only if a is positive. If a is negative, the inequality is reversed.

Given that $\displaystyle \frac{a}{b}> 0$ and a is positive then $\displaystyle \left(\frac{b}{a}\right)a= b> 0(a)= 0$ so b is also positive. But if a is negative then [tex]\left(\frac{a}{b}\right)a= b< 0(a)= 0[tex]. That is, if $\displaystyle \frac{b}{a}> 0$ then either a and b are both positive or a and b are both negative.
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5. ## Re: Positive a/b

Originally Posted by Plato
If $a=-4~\&~b=-3$ it is true that $\dfrac{-4}{-3}>0$ now you check out each in that list.
Check out each in the list by letting a = -4 and b = -3?

6. ## Re: Positive a/b

Originally Posted by harpazo
Check out each in the list by letting a = -4 and b = -3?
For example: E. says a+b>0.
If a=-4 & b=-3 then -4-3=-7 so E. is FALSE.
Do each of A B C D.

7. ## Re: Positive a/b

Originally Posted by Plato
For example: E. says a+b>0.
If a=-4 & b=-3 then -4-3=-7 so E. is FALSE.
Do each of A B C D.
a = -4, b = -3

ab > 0

(-4)(-3) > 0

12 > 0

True statement. Thus, C is correct.

8. ## Re: Positive a/b

Originally Posted by harpazo
a = -4, b = -3
ab > 0
(-4)(-3) > 0
12 > 0
True statement. Thus, C is correct.
Be careful The question asks which of the following must be true.
We found an example that makes E false so it is eliminated. the same also eliminated A , B, & D. BUT not C.
Counterexamples eliminate statements. BUT EXAMPLES CANNOT PROVE ANYTHING.
So to complete this question, you must explain why C must be true.

9. ## Re: Positive a/b

Originally Posted by Plato
Be careful The question asks which of the following must be true.
We found an example that makes E false so it is eliminated. the same also eliminated A , B, & D. BUT not C.
Counterexamples eliminate statements. BUT EXAMPLES CANNOT PROVE ANYTHING.
So to complete this question, you must explain why C must be true.
The answer is C because ab > 0 is a true statement given two negative numbers multiplied and two positive numbers multiplied.