Problem

How many integers are in the solution of the inequality |x+4|<9 ?

Answer

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Answer

Add the integers from both cases and the 0: 12+4+1=17.

Steps

Step 1 ：First, consider two cases: (1) x+4≥0 (or x≥−4), and (2) x+4<0 (or x<−4).

Step 2 ：For case (1), the inequality becomes x+4<9, which simplifies to x<5.

Step 3 ：For case (2), the inequality becomes −(x+4)<9, which simplifies to x+4>−9, and further to x>−13.

Step 4 ：Combining both cases, the solution is $-13

Step 5 ：The integers in this solution are -1 through -12 (12 integers), 1 through 4 (4 integers), and 0 (1 integer).

Step 6 ：Add the integers from both cases and the 0: 12+4+1=17.

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