Describe all the integer values of x that satisfy

abs(x)is less thatn abs(x+1)

To find the integer values of x that satisfy the inequality |x| < |x+1|, we need to break it down into two cases:

1. Case: x is positive
- Since x is positive, the absolute value of x is equal to x.
- Similarly, the absolute value of (x+1) is equal to (x+1).
- Therefore, the inequality becomes x < (x+1).
- Simplifying this further, we find that 0 < 1, which is always true.
- In this case, any positive integer value of x satisfies the inequality.

2. Case: x is negative
- Since x is negative, the absolute value of x is equal to -x.
- For (x+1), the absolute value is equal to -(x+1) = -x - 1, because it is a negative number.
- So, the inequality becomes -x < -x - 1.
- Adding x to both sides, we get 0 < -1.
- This is false for all negative numbers.
- Therefore, no negative integer values of x satisfy the inequality.

In conclusion, the integer values of x that satisfy the inequality |x| < |x+1| are all positive integers.