We are given the inequality:

\[
\frac{4}{2x - 1} \geq \frac{1}{x + 1}
\]

To solve this, let's follow these steps:

### Step 1: Eliminate denominators by multiplying both sides by \((2x - 1)(x + 1)\)
This approach is valid as long as we handle the conditions where \(x \neq \frac{1}{2}\) and \(x \neq -1\), since those values would make the denominators zero.

Multiplying both sides by \((2x - 1)(x + 1)\):

\[
(4)(x + 1) \geq (1)(2x - 1)
\]

### Step 2: Simplify both sides
Simplify the left and right sides:

\[
4(x + 1) = 4x + 4
\]
\[
2x - 1 = 2x - 1
\]

So the inequality becomes:

\[
4x + 4 \geq 2x - 1
\]

### Step 3: Solve for \(x\)
Now, let's solve for \(x\):

\[
4x + 4 \geq 2x - 1
\]

Subtract \(2x\) from both sides:

\[
2x + 4 \geq -1
\]

Subtract 4 from both sides:

\[
2x \geq -5
\]

Finally, divide by 2:

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

### Step 4: Consider restrictions
Recall that we must exclude the values \(x = \frac{1}{2}\) and \(x = -1\), because they would make the original denominators zero.

Thus, the solution is:

\[
x \geq -\frac{5}{2}, \quad \text{with} \quad x \neq \frac{1}{2} \quad \text{and} \quad x \neq -1
\]

### Final Answer:
The solution set is:

\[
x \in \left( -\frac{5}{2}, -1 \right) \cup \left( -1, \frac{1}{2} \right) \cup \left( \frac{1}{2}, \infty \right)
\]

Would you like further details or clarifications?

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Here are 5 related questions to consider:
1. How do we handle cases where multiplying by a variable expression might change the inequality sign?
2. What happens if the inequality has more complicated fractions?
3. Can this method be applied to inequalities with polynomials in the denominator?
4. How would the solution change if the inequality were strict (i.e., \(>\) instead of \(\geq\))?
5. How can graphing the solution set help in understanding the intervals?

**Tip:** Always check the domain of the original inequality to avoid solutions that make the denominator zero.

Explore how to solve the inequality 4/(2x - 1) ≥ 1/(x + 1) using rational expressions and inequalities. Learn the step-by-step process, including handling domain restrictions and simplifying expressions. Suitable for high school students in Grades 10-12.

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