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Mathematical Inconsistency?

Consider the following inequality

$$a-bc+dc>0$$

One can arrange it in the following way

Subtract both side by $$a$$

$$-bc+dc>-a$$

Multiply both side by $$-1$$

$$bc-dc<a$$

Factorize $$c$$ on LHS and divide both side by $$(b-d)$$ and we end up with

$$\boxed{c<\frac{a}{b-d}}$$

Now consider the original inequality, $$a-bc+dc>0$$ again. We can also rearrange it in the following way:

Again, subtract both side by $$a$$

$$-bc+dc>-a$$

Now instead of multiplying both side by $$-1$$, we factorize $$c$$ on left hand side and divide both side by $$(-b+d)$$ and we can obtain the following result

$$c>\frac{-a}{-b+d}$$

= $$\boxed{c>\frac{a}{b-d}}$$

It seems that the first and second result contradicts each other, so what is going on? Was there any flaw in the rearrangement?

Note by Tan Wei Xin
2 years, 4 months ago

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In the first result, you divided both sides by $$b-d$$, while in the second, you divided by $$d-b$$. One of the differences must be negative, so you need to reverse the inequality sign when dividing.

- 2 years, 4 months ago

Nice observation. Thanks!

- 2 years, 4 months ago

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