Find the sum of all integers that satisfy these conditions: \[
|x|+1>7\text{ and }|x+1|\le7.
\]
To find the sum of all integers that satisfy the given conditions, I need to find out what those integers are first. I can start by solving the inequalities separately. For the first inequality, I can subtract 1 from both sides to get \[
|x|>6.
\] This means that either x is greater than 6, or x is less than -6. For the second inequality, I can use the definition of absolute value to write it as two cases: \[
x+1\le7\text{ or }x+1\ge-7.
\] Simplifying these, I get \[
x\le6\text{ or }x\ge-8.
\] Now I can combine the two inequalities to find the intersection of their solutions. I need x to be both greater than 6 and less than or equal to 6, or both less than -6 and greater than or equal to -8. The first case is impossible, since there is no number that is both greater than 6 and less than or equal to 6. The second case gives me two possible values for x: -7 and -8. These are the only two integers that satisfy both inequalities. To find their sum, I just add them together: \[
(-7)+(-8)=-15.
\]

