|2x+1| ≤4

To solve the inequality \(|2x + 1| \leq 4\), we need to consider the definition of absolute value. The inequality can be split into two separate inequalities:

1. \(2x + 1 \leq 4\)
2. \(2x + 1 \geq -4\)

Now, let's solve each inequality separately.

For the first inequality:

\[ 2x + 1 \leq 4 \] Subtract 1 from both sides: \[ 2x \leq 3 \] Now divide by 2: \[ x \leq \frac{3}{2} \]

For the second inequality:

\[ 2x + 1 \geq -4 \] Subtract 1 from both sides: \[ 2x \geq -5 \] Now divide by 2: \[ x \geq -\frac{5}{2} \]

Combine the results:

Combining both inequalities, we get: \[ -\frac{5}{2} \leq x \leq \frac{3}{2} \]

Conclusion:

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