WholeLottaLove wrote:
Thanks!
I understand the bit about plugging in values, but what I don't understand is the explanation given in the book. What I don't understand is why it's [- infinity to 3] and [7 to infinity] in the inequality we have (x-7)(x-3)>/=0 so wouldn't we test for all cases greater than or equal to 3 and greater than or equal to 7? Why does the number line have x is less than or equal to 3 (i.e. [-infinity to 3]?
I understand the bit about plugging in values, but what I don't understand is the explanation given in the book. What I don't understand is why it's [- infinity to 3] and [7 to infinity] in the inequality we have (x-7)(x-3)>/=0 so wouldn't we test for all cases greater than or equal to 3 and greater than or equal to 7? Why does the number line have x is less than or equal to 3 (i.e. [-infinity to 3]?
If you'd look at the graph I plotted for the function, you'd realize that for [- infinity to 3] and [7 to infinity], the function satisfies the condition of being > 0 i.e. our Statement 2. The values 3 and 7 lie on the function where the value is 0. If we were to use any value greater than equal to 3 but less than 7, the function will no longer be > 0. For such values between {3,7} the values of f(x) < 0. Hence we cannot use them as valid values of x satisfying the inequality!
ee <---------- eeeeeeeeeeeee ----------->
|__________3___________7___________|
Now if you look at the number line mentioned in the book, for all values less than 3, f(x) > 0. Similarly for all values of x > 7, f(x) > 0. Now since for the region x < 3, we have values like {-1,-2.....} -ve values for x. We can not use the same as a solution for x>0, our question! This is the reason why the above number line has arrows starting from 3 towards -ve infinity and 7 to positive infinity to represent the solution of f(x) > 0.
Hope this answers your question!
Regards,
Arpan