Solve the following

Question

Solve the following quadratic inequality algebraically, and write the answer in interval notation:

 

(x+4)(x-4) greater than -6x

Answer 1

(x+4)(x-4)>-6x

x²-16>-6x

x²-16+6x>0

x²+6x-16>0

(x+8)(x-2)>0

there are two conditions...

 (x+8)>0 and (x-2)>0 or (x+8)<0 and (x-2)<0

first condition...

(x+8)>0 and (x-2)>0

x>-8 and x>2

for these two conditions to be true at the same time, x>2 (x can't just be greater than -8 because there are numbers greater than -8 that are not greater than 2)

look at it this way: you want the intersection of these two sets of numbers

we are using the set of Real numbers but assume we are using the set of Integers

{-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,...}

{3,4,5,6,7,8,9,...}

what numbers do these sets have in common ? {3,4,5,6,7,8,9,...}, in other words x>2

on to the next condition...

(x+8)<0 and (x-2)<0

x<-8 and x<2

for these two conditions to be true at the same time, x<-8 (x can't just be less than 2 because there are numbers less than 2 that are not less than -8)

the answer is {x: x<-8 or x>2} and remember we are using the set of Real numbers, not just Integers; the integers were used only for illustrative purposes