Question 1141453
<br>
One way to solve absolute value problems like this is to interpret<br>
{{{abs(x-a)<=b}}}<br>
as meaning the difference between x and a is at most b.  So<br>
{{{abs(x-68)<=9}}}<br>
means x is at most 9 away from 68.  68-9 = 59; 68+9 = 77.  The number of integers from 59 to 77 inclusive is (77-59)+1 = 19.<br>
A)  Using the same interpretation, x has to be less than 8 away from -4.  -4-8 = -12; -4+8 = 4.  Since this is a strict inequality, the integer solutions are from -11 to 3 inclusive, which is 15 values.<br>
B)  This one says x is less than 3 away from 5: from 3 to 7, which is 5 values.<br>
From these three problems, you should note that<br>
(1) the number of integer solutions to {{{abs(x-a)<=b}}} is 2b+1; and
(2) the number of integer solutions to{{{abs(x-a)<b}}} is 2b-1.