Help, what are some practical examples for multiplying negatives

Question

It seems like a huge swing from positive to negative then back to positive. What are some real world applications for multiplying negatives

Answer 1

I make five withdrawals (that's a negative) of $10 each. 

What is the result? 5(-10) = -50.

 

A scuba diver descends (that's a negative) by 10 foot increments. He does this three times. How far did he go?

3(-10 feet) = -30 feet.

 

On each of three plays a football team lost (that's negative) 2 yards. Where are they now?

3(-2 yds) = -6 yards

Answer 2

you are right about swinging from negative to positive; in general, the opposite of x is -x (this is a definition). Whatever x is, on the number-line, to get its opposite, you imagine going along a semicircle centered at the origin (0) passing through the number x...go along the semicircle to the opposite side of the origin, and you arrive at -x.

 

We summarize this by such things as -(3) = -3, i.e. the opposite of 3 is "negative three".

similarly, the opposite of negative three is 3: -(3) = 3.

This is generalized to say "a negative times a negative is a positive."

 

An example in algebra: 3x + 5 = 4x - 7

Then we add the opposite of 3x to each side:  3x +(-3x) + 5 = 4x + (-3x) -7

and that simplifies to 5 = x -7 and finally we add 7 to each side, concluding with 5 + 7 = x so x = 13.

 

This can be shortened to the TRANSPOSING A TERM principle: A term can move to the opposite side of an equation or inequality, where it takes on the opposite sign. 

 

An example: Solve (-½)x = -3

Multiply both sides by -2 (which is the negative reciprocal of the coefficient of x; you get

(-2)(-½)x = (-2)(-3)  or  x = 6.

This answer checks.  And you see how a negative times a negative occurs in an application.