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Let's address the two parts of this problem step by step:

### a. Identify the value(s) of the variable that make the denominator zero.

The denominators in the equation are both \( x + 1 \). To find the restriction on the variable \( x \), we solve the equation that makes the denominator zero:

\[
x + 1 = 0
\]

Solving for \( x \):

\[
x = -1
\]

Thus, \( x = -1 \) is a restriction on the variable, meaning \( x = -1 \) would make the denominator zero, which is not allowed.

### b. Solve the equation

The given equation is:

\[
\frac{1}{x+1} + 2 = \frac{5}{x+1}
\]

To solve this, we first subtract \(\frac{1}{x+1}\) from both sides:

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

Simplifying the right-hand side:

\[
2 = \frac{5 - 1}{x+1} = \frac{4}{x+1}
\]

Now, multiply both sides by \(x+1\) to eliminate the denominator (but keep in mind the restriction that \(x \neq -1\)):

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

Expanding:

\[
2x + 2 = 4
\]

Subtract 2 from both sides:

\[
2x = 2
\]

Now, divide by 2:

\[
x = 1
\]

Thus, the solution is \( x = 1 \).

Since \( x = 1 \) does not violate the restriction \( x \neq -1 \), this solution is valid.

### Final Answer for Part b:
The solution set is \( \{ 1 \} \).

So, the correct choice is **A**.

---

Do you have any questions or need further details?  
Here are some related questions you might find helpful:
1. What happens if the restriction \(x = -1\) is ignored in rational equations?
2. How do you solve rational equations with different denominators?
3. What are the steps to clear fractions in a rational equation?
4. How do restrictions on variables affect the solution set in rational equations?
5. Why is it important to check the restrictions after solving rational equations?

**Tip:** Always check your solution against any restrictions to ensure the values you find are valid solutions!

The following rational equation has denominators that contain variables. For this equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 

Equation: 1 / (x + 1) + 2 = 5 / (x + 1)

Learn how to solve the rational equation 1/(x + 1) + 2 = 5/(x + 1) by identifying variable restrictions and clearing denominators. This step-by-step solution covers key algebraic concepts like rational equations and variable restrictions. Suitable for students in Grades 9-12.

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