Question 1209263: a, b and c are integers. If a < b, then ac < bc. Is the statement true or false?
Answer by math_tutor2020(3817)   (Show Source):
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Answer: False
Explanation
We only need one counter-example to disprove this claim.
This is because the claim encompasses all integers (and we can change "integers" to "real numbers" to get the same idea).
Let's say that,
a = 1, b = 2, c = -3
a < b is true since 1 < 2 is true.
However,
a*c < b*c
1*(-3) < 2*(-3)
-3 < -6
is false
Use a number line to see that -6 is to the left of -3, so -6 is actually the smaller item.
A vertical number line could help paint a better picture. The -6 is further down compared to -3.
How to make the statement true? Simply mention "c is positive".
Incorrect: "a < b leads to ac < bc"
Correct: "If c is positive, then a < b leads to ac < bc"
Or if you said "if c is negative, then a < b leads to ac > bc", it would be a true claim.
I've seen many students forget to flip the inequality sign when multiplying both sides by a negative number.
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