For what values of x is the linear approximation sqrt(x+3) ~ 7/4 + x/4

accurate to within 0.5?

"~" stands for about

Solution: Accuracy to within 0.5 means that the function should differ by less than 0.5:

|sqrt(x+3) - ( 7/4 + x/4) | > 0.5

Equivalently we could write

sqrt(x+3) - 0.5 < ( 7/4 + x/4) < sqrt(x+3) + 0.5

I have two questions:

First, how do they get from "linear approximation sqrt(x+3) ~ 7/4 + x/4 accurate to within 0.5" to the expression "|sqrt(x+3) - ( 7/4 + x/4) | > 0.5"

Second, how do they get the equvalent term from

|sqrt(x+3) - ( 7/4 + x/4) | > 0.5

to

sqrt(x+3) - 0.5 < ( 7/4 + x/4) < sqrt(x+3) + 0.5

I thought |u| < a --> -a < u < a

So, I get -0.5 < sqrt(x+3) - ( 7/4 + x/4) < 0.5

What do I have to do to get to

sqrt(x+3) - 0.5 < ( 7/4 + x/4) < sqrt(x+3) + 0.5

Thanks for any help!

Additionally I want to add this information I have:

F(x)= sqrt(x+3)

Linear approximation f(x) ~ f(a)+f'(a)(x-a)

a=1