Explain how solving -7y > 161 is different from solving 7y > -161.

Explanation:

The method of solution is substantially the same, but there is a detail that needs to be attended to.

If the solution method in the first case is to divide by the coefficient of y, then that division (or multiplication) by a negative number changes the ordering. The ">" symbol needs to be reversed to a "<" symbol.

  -7y > 161

  y < -23 . . . . . . after division by -7

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In the second case, dividing by the coefficient of y gives ...

  7y > -161

  y > 23 . . . . . . . no reordering necessary for division by a positive number

Alternate method for dealing with negative coefficients

The equation in the first case can also be solved by adding 7y-161 to get ...

  -161 > 7y

Now, division by 7 proceeds the same as in the second case, with no reordering required:

  -23 > y

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Personal note

I was completely mystified by the solution of inequalities until I realized this rule. Multiplication or division by a negative number reverses the ordering.

There are also some other operations that can be performed on inequalities that reverse the ordering. A simple example is taking the reciprocal.

For an equation in which a=b, it is also true that 1/a = 1/b. For an inequality, ...

  a > b

  1/a < 1/b . . . . . when a and b have the same sign, the ordering symbol is reversed

Actually, any function in which the slope is negative in the relevant domain can require the ordering to be reversed. Trig functions can do that: for first and second quadrant angles, a > b means cos(a) < cos(b).

What it amounts to is that you need to pay attention to ordering if you're doing anything other than addition and subtraction and multiplication by a positive number.

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step-by-step explanation:

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