"or" means Union; symbol: ∪ (ex. either x < 2 or x > 5)

"and" means Intersection; symbol: ∩ (occurs in both sets at the same time) may be written 2 < x < 3 (x is greater than 2 and less than 3)

|xy| = |x| ×| y| , but |x| + |y| is not necessarily equal to |x + y|

|x - 2| = |2 - x|, although usually x - 2 ≠ 2 - x

I. PRODUCTS AND QUOTIENTS

However, in (1), (2) and (3) above, we may multiply both sides by (-1) (reversing the inequality sign):
that is:

(x - 2)(4 - x) > 0   ⇒   (x - 2)(4 - x)(-1) < 0(-1)   ⇒   (x - 2)(x - 4) < 0
and
< 4   ⇒   (-1) > (-1)(4)   ⇒   > -4
and
< 0   ⇒   > (-1)(0)   ⇒   > 0

Also, in (3) we may multiply numerator and denominator by (-1). < 0   ⇒ < 0

II. Examine the following:

a) (x - 2)(x + 3) = 0 says "x - 2 = 0 OR x + 3 = 0"
b) (x - 4)(x + 1) < 0 does not say "x - 4 < 0 OR x + 1 < 0"
c) (x + 3)(x - 4) > 0 does not say "x + 3 > 0 OR x - 4 > 0"
d) (x + 3)(x + 1) = -1 does not say "x + 3 = -1 OR x + 1 = -1"

III. Describe each of the following. Then solve for x using the number line.

a) < 0                    b) < 1              c) < 0

d) > 0                     e) < 0

IV. Describe each of the following on the number line ("distance" method).  Write each without absolute value signs.

a)   |x| = 2       b)   |x| < 2      c)   |x - 3| = 2           d)   |x - 3| = -2

e)   |x - 3| ≥ 2      f)   |x - 3| < 2      g)   |3x - 8| < 7

Answers:

III.  a) -1 < x < 2      b) x > -2       c) 0 < x < 2         d) x > 1         e) -1 < x < 2 or x > 4

IV.  a) x = 2, -2     b) -2 < x < 2      c) x = 1, 5    d) ∅       e) x ≤ 1 or x ≥ 5    f) 1 < x < 5    g) < x < 5