|2x+1|_<4

To solve the inequality \(|2x + 1| \leq 4\), we will break it down into two cases based on the definition of absolute value.

The expression \(|A| \leq B\) means that \(-B \leq A \leq B\). Applying this to our inequality:

1. Set up the inequality: \[ -4 \leq 2x + 1 \leq 4 \]

2. We will split this into two parts:

   First part: \[ -4 \leq 2x + 1 \]

   Subtract 1 from both sides: \[ -4 - 1 \leq 2x \] \[ -5 \leq 2x \]

   Divide by 2: \[ -\frac{5}{2} \leq x \]

   Second part: \[ 2x + 1 \leq 4 \]

   Subtract 1 from both sides: \[ 2x \leq 4 - 1 \] \[ 2x \leq 3 \]

   Divide by 2: \[ x \leq \frac{3}{2} \]

3. Now, combine the results from both parts: \[ -\frac{5}{2} \leq x \leq \frac{3}{2} \]

Thus, the solution to the inequality \(|2x + 1| \leq 4\) is: \[ x \in \left[-\frac{5}{2}, \frac{3}{2}\right] \]