|2x+1| ≤4

To solve the inequality \(|2x + 1| \leq 4\), we need to consider the definition of absolute value. The expression \(|A| \leq B\) implies that \(-B \leq A \leq B\).

So, we can split the inequality into two parts:

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

Now, let's solve each part 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:

\[ x \geq -\frac{5}{2} \]

Now, we can combine the results:

\[ -\frac{5}{2} \leq x \leq \frac{3}{2} \]

Thus, the solution to the inequality \(|2x + 1| \leq 4\) is:

\[ x \in \left[-\frac{5}{2}, \frac{3}{2}\right] \]