Multplying by a negative and inequalities

Question

I know that if you multiply or divide a negative across a inequality you have to reverse the sign. What about is you cross multiply across an inequality when both the numerator and denominator are negative. Then do you flip the sign twice to get back to the orginal symbol?

crack700 · Answer

Yes, the sign will become the original sign if you cross multiply denominators which are both negative.
For example 
 \frac{1}{-2} < \frac{1}{-3} 
If you cross multiply sign will remain constant -3 < -2

Jinglander · Answer

And if both positives are on the same side then you should make it positive. Also what about with varibles

IanStewart · Answer

When you cross multiply, what you're actually doing is multiplying on both sides by both denominators. So if you cross multiply in an inequality, you certainly need to know the signs of both denominators. If both are negative, you're really multiplying by two negatives, so you're perfectly correct - the inequality would not change direction. But, if exactly one of the two denominators was negative, you'd need to reverse the inequality if you cross multiply. Be especially careful if your denominators are unknowns, of course - you'd need to know whether your unknowns were positive or negative in order to proceed.

brau0300 · Answer

Yes, remember that cross multiplication is really a series of single multiplications.

e.g. cross multiplying the following inequality could be viewed as a multi step process

\frac{-1}{3}\frac{-3}{4}

Step 2: Multiply Both Sides by (-4), Flip Inequality Direction

\frac{-4*3}{3}<\frac{3*4}{4}

Step 3: Reduce

-4<3

Notice that if we had elected to multiply both sides by positive 4 or 3 instead of -4 and -3, we would still have a valid inequality. Viewing the cross multiplication as individual multiplications allows you to apply the simple rule: Only flip the inequality direction when you multiply both sides by a single negative number.

Multplying by a negative and inequalities

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