Explain the rule in solving inequalities with a negative coefficient.

  the comparison must be reversed when multiplying by a negative

Step-by-step explanation:

The rules of equality apply to solving inequalities, with the exception that multiplication or division by a negative number reverses the sense of the comparison:

  -x > 1

  x < -1 . . . . . multiply both sides by -1

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Effectively, multiplication (or division) by a negative number is equivalent to reflection across the origin. Things that were ordered left/right (on the number line) are ordered right/left after such a reflection:

  -2 < -1

  1 < 2

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Additional comment

Application of any function requires that you pay attention to ordering. Some functions naturally reverse the order; others do so only on specific domains.

Consider f(x) = 1/x.

  1 < 2

  f(1) > f(2) . . . . because the slope of the f(x) function is negative everywhere.

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In the first and second quadrants, the cosine function also reverses order.

  20° > 10°

  cos(20°) < cos(10°)

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Probably the most commonly encountered function used with inequalities is the absolute value function.

  |x-3| > 2

This function has one domain where the slope is negative, and another domain where the slope is positive.

  For x < 3, the function negates its argument, so we have ...

     -(x -3) > 2

     -x +3 > 2

     -x > -1

     x < 1 . . . . . everywhere consistent with x < 3

  For x ≥ 3, the function does nothing, so we have ...

     x -3 > 2

     x > 5

  The solution to this absolute value inequality is (x < 1) ∪ (x > 5).

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You can also resolve negative coefficients by adding the opposite. Addition and subtraction never require any change to the comparison operator.

  -x > 1

  0 > x +1 . . . . x is added

  -1 > x . . . . . . -1 is added