Question

HELP PLEASE
|x +2| < |x|

Answer 1

To solve this inequality, we need to consider the different cases where the absolute value of x + 2 is less than the absolute value of x.

If x is positive, then |x| is equal to x, and we can rewrite the inequality as follows:

|x + 2| < x

Then we can subtract x from both sides of the inequality to get:

|x + 2| - x < 0

We can simplify the left side of the inequality as follows:

|x + 2 - x| < 0

This simplifies to:

|2| < 0

Since the absolute value of 2 is not less than 0, this inequality has no solution in the case where x is positive.

If x is negative, then |x| is equal to -x, and we can rewrite the inequality as follows:

|x + 2| < -x

Then we can add x to both sides of the inequality to get:

|x + 2| + x < 0

We can simplify the left side of the inequality as follows:

|x + 2 + x| < 0

This simplifies to:

|2x + 2| < 0

This inequality has no solution in the case where x is negative.

Therefore, the solution to the inequality |x + 2| < |x| is the empty set, which means that there are no values of x that satisfy the inequality.