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# Algebra/what is a counterexample

Question
Hello,Scott!

Can you clear up this one that confuses me?

"Give a counterexample to show that the following statement is not always true for real numbers a, b, and c:
If a>b, then ac>bc"

A.a=3,b=4,c=2     3(2) is not greater than or equal to 4(2)

B.a=-3,b=4,c=2   -3(2) is greater than or equal to 4(2)

C.a=3,b=-4,c=2    3(2) is greater than or equal to -4(2)

D.a=3,b=4,c=-2    3(-2) is not greater than or equal to 4(-2)

The correct solution is given as a=3, b=4, c=-2, 3(-2)is not greater than or equal to 4(-2)

I don't get it! What the h___  is a counterxample?
Why is choice D. the correct answer?

This sounds like the kind of stuff that lawyers love.

Thanks

Chris

Upon getting this question, I too am confused.  Send this question to the publisher of the book, for the correct answer is not found.  It is known that with the thousands of problems in a book, an error is almost bound to occur somewhere.

A true counterexample would be to find a, b such that a>b with a value for c that makes ac<=bc.

For (A), a=3 is not greater than b=4, so it can't be used to show anything.

For (B), a=-3 is not greater than b=4, so it can't be used to show anything.

For (C), a=3 is greater than b=-4, but 3*2 = 6 is also greater than -4*2 = -8,
so this doesn't show anything is wrong.

For (D), a=3 is not greater than b=4, so it can't be used to show anything.

Perhaps the negative sign needs to be moved on (D).
I would use the number a=4, b=3, and c=-2.
That would say, using (D), since 4 > 3, deducing ac > bc does not work.
When, from what's given for values of a, b, and c, ac = -12 and bc = -6,
It can then be seen that ac is not greater than bc.
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment This guy knows his stuff! Congrats! Chris

Algebra

Volunteer

#### Scott A Wilson

##### Expertise

Any algebraic question you've got. That includes question that are linear, quadratic, exponential, etc.

##### Experience

I have solved story problems, linear equations, parabolic equations. I have also solved some 3rd order equations and equations with multiple variables.

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Documents at Boeing in assistance on the manufacturiing floor.

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MS at math OSU in mathematics at OSU, 1986. BS at OSU in mathematical sciences (math, statistics, computer science), 1984.

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Both my BS and MS degrees were given with honors.

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Students in a wide variety of areas since the 80's; over 1,000 of them have been in algebra.