Solve the inequality x^2 - 10x + 30 > 9, and express the answer in interval notation

Question

Solve the inequality x^2 - 10x + 30 > 9, and express the answer in interval notation

Answer 1

 

x²-10x+30 > 9             subtract 9 from both sides

 

x2-10x+21 > 0             factor the trinomial:      factors of 21 that add to -10:   -7 and -3

 

(x-7)(x-3) >0               gives x = 7 and x = 3

 

 

These two values break the number line into 3 sections:

 

_______i___________i_________

              3                     7                   

Test one value in each section in the inequality (x-7)(x-3) >0 to see which sections give a true answer.

 

Test 0:    (0-7)(0-3)>0    -7(-3) = 21 > 0 TRUE

Test 5:    (5-7)(5-3)>0     -2(2) = -4  > 0   False

Test 8:    (8-7)(8-3)>0     1(2) = 2  > 0   True

 

The answer is   (-∞,3) ∪ (7,∞)