Find all numbers x for which i x1x30 When is a product of 2

Find all numbers x for which,

(i) (x-1)(x-3)>0. (When is a product of 2 numbers positive?)

(ii) Ix-3I<8.

Solution

Sol(1)

(x-1)(x-3)>0.

(x-1)>0 and x-3>0

x>1 and x>3

Solution:

x<1 or x>3

Interval notation:(-,1) U(3,)

Solution2:

Ix-3I<8

|f(x)| < a implies f(x)<a and f(x)>-a

x-3<8 and x-3>-8

x-3<8:x<8+3:x<11

x-3>-8: x>-8+3:x>-5

combine the ranges

solution:

-5<x<11

interval notation:

(-5,11)