# how to find a and b when there is absolute value inequalities

I was given a number line which cords were $$x>5$$ and $$x<-7$$, and I was given this equation $$|x-a|>b$$. I have no idea how to find the value of $$a$$ and $$b$$. I tried different ways of solving this problem but I don't think any of them meets the original solutions.

• This is very similar to your prior question Why not take a look at the answers you received to that question and see if the same techniques apply here? – lulu Feb 10 at 18:18

Hint: You have to consider the cases $$x\geq a$$ then we will get $$x>a+b$$ and $$x then we will have $$-x+a>b$$
If you have that $$x < -7$$ and $$x > 5$$, try and find the middle point between these two: $$x_\text{mid} = {-7 + 5 \over 2} = {-2\over 2} = -1.$$ This gives you the $$a$$ you seek. To find the $$b$$, how far away is $$5$$ or $$-7$$ from $$-1$$? Well, $$5 - (-1) = 5+1 = 6.$$ Thus, the radius of the interval is $$6$$ units. This gives you the $$b$$ you want. Hence, $$|x - (-1)| > 6 \quad\implies\quad|x+1| > 6 \quad\implies\quad x + 1 > 6 \text{ or } x + 1 < -6.$$ Thus, $$x > 5 \text{ or } x < -7,$$ as desired.