|2X+3|<1

To solve the inequality \( |2X + 3| < 1 \), we need to break it down into two separate inequalities. The expression \( |a| < b \) implies that \( -b < a < b \).

So, we can rewrite the inequality as follows:

\[ -1 < 2X + 3 < 1 \]

Now, we will split this into two parts:

1. First part: \[ -1 < 2X + 3 \]

   Subtract 3 from both sides: \[ -1 - 3 < 2X \] \[ -4 < 2X \]

   Now, divide by 2: \[ -2 < X \quad \text{or} \quad X > -2 \]

2. Second part: \[ 2X + 3 < 1 \]

   Subtract 3 from both sides: \[ 2X < 1 - 3 \] \[ 2X < -2 \]

   Now, divide by 2: \[ X < -1 \]

Combining the results from both parts, we get:

\[ -2 < X < -1 \]

Thus the solution to the inequality \( |2X + 3| < 1 \) is:

\[ X \in (-2, -1) \]