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.

Answer 2

To solve the inequality |x + 2| < |x|, we can break it down into two cases based on the absolute value.

Case 1: x + 2 ≥ 0 (x is greater than or equal to -2) In this case, the absolute value of x + 2 is equal to x + 2. Similarly, the absolute value of x is also equal to x. So we can rewrite the inequality as follows: x + 2 < x

Now, let's solve the inequality: x + 2 < x Subtracting x from both sides: 2 < 0

Since the inequality 2 < 0 is false, there are no solutions in this case.

Case 2: x + 2 < 0 (x is less than -2) In this case, the absolute value of x + 2 is equal to -(x + 2) because x + 2 is negative. The absolute value of x is still equal to -x. So we can rewrite the inequality as follows: -(x + 2) < -x

Now, let's solve the inequality: -(x + 2) < -x Distributing the negative sign: -x - 2 < -x

The -x terms cancel out: -2 < 0

Since the inequality -2 < 0 is true, all values of x that are less than -2 satisfy the inequality.

Therefore, the solution to the inequality |x + 2| < |x| is x < -2.