The rule for multplying and dividing both sides of an inequality do not meantion zero. Explain why??
In general, we do not multiply or divide each side of an equality by zero, as it could give erroneous results. So the same applies to inequalities.
For example:
A=B
Multiply by A:
A*A = A*B
Subtract B²:
A²-B² = AB - B²
Factor each side:
(A+B)(A-B) = B(A-B)
Cancel out (A-B)
A+B = B
How does the fallacy work: we divided each side by (A-B) which is zero!
The omission of zero in the rules for multiplying and dividing both sides of an inequality is due to the fact that dividing by zero is undefined in mathematics. When we set up an inequality, we are essentially comparing two quantities that may have different values. We use symbols like "<" (less than) or ">" (greater than) to represent these comparisons.
When we multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains unchanged. For example, if we have the inequality "a < b", and we multiply both sides by a positive number "c", we get "ac < bc". Similarly, if we divide both sides by a positive number "c", the inequality still holds.
However, if we were to multiply or divide both sides of an inequality by zero, it would lead to inconsistency and contradiction. Division by zero is undefined, meaning there is no mathematical operation to assign a result. Furthermore, any equation or inequality involving division by zero would lead to incorrect or nonsensical conclusions.
To summarize, the rules for multiplying and dividing both sides of an inequality do not mention zero because division by zero is undefined and leads to inconsistent and contradictory results in mathematics.