\frac{1}{2}\left|2x+1\right|=\left|-4+1\right|

Let's solve the equation step by step:

The equation is:

\[
\frac{1}{2}\left|2x+1\right|=\left|-4+1\right|
\]

1. **Simplify the absolute value on the right side**:

\[
\left|-4 + 1\right| = \left|-3\right| = 3
\]

So the equation becomes:

\[
\frac{1}{2}\left|2x + 1\right| = 3
\]

2. **Multiply both sides by 2** to eliminate the fraction:

\[
\left|2x + 1\right| = 6
\]

3. **Solve the absolute value equation**:

The equation \(\left|2x + 1\right| = 6\) can be split into two cases:

**Case 1: \(2x + 1 = 6\)**

\[
2x = 6 - 1
\]
\[
2x = 5
\]
\[
x = \frac{5}{2}
\]

**Case 2: \(2x + 1 = -6\)**

\[
2x = -6 - 1
\]
\[
2x = -7
\]
\[
x = \frac{-7}{2}
\]

Thus, the solutions are:

\[
x = \frac{5}{2} \quad \text{or} \quad x = \frac{-7}{2}
\]

### Final Answer:
\[
x = \frac{5}{2} \quad \text{or} \quad x = \frac{-7}{2}
\]

Would you like more details on any step or have any questions?

### Related Questions:
1. How do you solve absolute value inequalities?
2. What is the difference between solving absolute value equations and inequalities?
3. Can absolute value equations have more than two solutions?
4. How does multiplying both sides of an equation by a fraction affect the solution?
5. What happens if there is no solution to an absolute value equation?

**Tip:** When dealing with absolute value equations, always consider both the positive and negative scenarios to capture all possible solutions.

Learn how to solve the absolute value equation \frac{1}{2}\left|2x+1\right|=\left|-4+1\right| with a detailed step-by-step solution. This problem covers key algebraic concepts, including the absolute value theorem and solving equations with absolute values. Suitable for students in Grades 9-11.

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