The inequality you provided is:

$\frac{4}{2x - 1} \geq \frac{1}{x + 1}$

To solve for $x$, follow these steps:

Step 1: Cross-multiply

First, we eliminate the denominators by cross-multiplying, but we must take care that the denominators are non-zero. We'll discuss the restrictions later. The cross-multiplication gives:

$4(x + 1) \geq 1(2x - 1)$

Step 2: Expand both sides

Expand both sides of the inequality:

$4x + 4 \geq 2x - 1$

Step 3: Isolate $x$

Now, move all the terms involving $x$ to one side, and constants to the other:

$4x - 2x \geq -1 - 4$

This simplifies to:

$2x \geq -5$

Step 4: Solve for $x$

Now, divide both sides by 2:

$x \geq -\frac{5}{2}$

Step 5: Check the restrictions

To ensure the solution is valid, we need to check that the original denominators are not zero. This gives two conditions:

1. $2x - 1 \neq 0$ implies $x \neq \frac{1}{2}$
2. $x + 1 \neq 0$ implies $x \neq -1$

Final Answer

The solution is $x \geq -\frac{5}{2}$, but we must also exclude $x = \frac{1}{2}$ and $x = -1$. Therefore, the solution set is:

$x \in \left[ -\frac{5}{2}, \infty \right) \setminus \left\{ -1, \frac{1}{2} \right\}$

Would you like more details or have any questions?

Related Questions

1. How do you determine the domain of rational inequalities?
2. What happens if the inequality was strict (e.g., $>$ instead of $\geq$)?
3. Can we use graphing techniques to solve rational inequalities?
4. What is the importance of cross-multiplication in inequalities?
5. How do vertical asymptotes relate to restrictions on $x$?

Tip

Always check for restrictions from denominators in rational equations to avoid invalid solutions.