17/3 divided by 2/3 help please!:))

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

____

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

_____

<em>Additional comment</em>

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.

__

<em>In the first and second quadrants</em>, the cosine function also reverses order.

20° > 10°

cos(20°) < cos(10°)

__

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).

__

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

7 0

3 years ago