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

Question

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.

Answer 1

Hint: You have to consider the cases $$x\geq a$$ then we will get $$x>a+b$$ and $$x<a$$ then we will have $$-x+a>b$$

Answer 2

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.