Solve these algebraic equations and state the restrictions for each solution.

We are given 10 equations to solve. The user has circled a), c), and g) — so I will solve those three, as requested, and explain each step carefully.

---

## ✔ Problem a)
Solve:
\[
\frac{x}{2} + 2 = \frac{3}{5}
\]

Step 1: Subtract 2 from both sides

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

Convert 2 to fifths: \(2 = \frac{10}{5}\)

\[
\frac{x}{2} = \frac{3}{5} - \frac{10}{5} = -\frac{7}{5}
\]

Step 2: Multiply both sides by 2

\[
x = 2 \cdot \left(-\frac{7}{5}\right) = -\frac{14}{5}
\]

✔ Answer:
\[
\boxed{x = -\frac{14}{5}}
\]

Restrictions? None — this is a linear equation with no denominators involving variables. So no restrictions.

---

## ✔ Problem c)
Solve:
\[
\frac{2}{7} - \frac{k - 4}{3} - \frac{k}{21} = 1
\]

Step 1: Find the least common denominator (LCD)

Denominators: 7, 3, 21 → LCD = 21

Multiply every term by 21 to eliminate denominators:

\[
21 \cdot \left( \frac{2}{7} \right) - 21 \cdot \left( \frac{k - 4}{3} \right) - 21 \cdot \left( \frac{k}{21} \right) = 21 \cdot 1
\]

Simplify each term:

- \(21 \cdot \frac{2}{7} = 6\)
- \(21 \cdot \frac{k - 4}{3} = 7(k - 4)\)
- \(21 \cdot \frac{k}{21} = k\)
- Right side: 21

So we get:

\[
6 - 7(k - 4) - k = 21
\]

Step 2: Distribute and simplify

\[
6 - 7k + 28 - k = 21
\]
\[
(6 + 28) + (-7k - k) = 21
\]
\[
34 - 8k = 21
\]

Step 3: Solve for k

Subtract 34 from both sides:

\[
-8k = 21 - 34 = -13
\]

Divide by -8:

\[
k = \frac{-13}{-8} = \frac{13}{8}
\]

✔ Answer:
\[
\boxed{k = \frac{13}{8}}
\]

Restrictions? None — all denominators are constants, so no variable in denominator. No restrictions.

---

## ✔ Problem g)
Solve:
\[
\frac{x + 1}{x - 2} + 2 = \frac{9}{x - 2}
\]

Step 1: Note restrictions

Since there are denominators with \(x - 2\), we must have:

\[
x - 2 \ne 0 \Rightarrow x \ne 2
\]

✔ Restriction: \(x \ne 2\)

Step 2: Eliminate denominators

Multiply every term by \(x - 2\) (the common denominator):

\[
(x - 2) \cdot \left( \frac{x + 1}{x - 2} \right) + (x - 2) \cdot 2 = (x - 2) \cdot \left( \frac{9}{x - 2} \right)
\]

Simplify:

- First term: \(x + 1\)
- Second term: \(2(x - 2)\)
- Third term: 9

So:

\[
x + 1 + 2(x - 2) = 9
\]

Step 3: Expand and simplify

\[
x + 1 + 2x - 4 = 9
\]
\[
3x - 3 = 9
\]

Step 4: Solve for x

Add 3 to both sides:

\[
3x = 12
\]

Divide by 3:

\[
x = 4
\]

Step 5: Check restriction

We had \(x \ne 2\), and \(x = 4\) is allowed.

✔ Answer:
\[
\boxed{x = 4}
\]

---

## 📌 Final Answers for Circled Problems:

- a) \(\boxed{x = -\dfrac{14}{5}}\) — *No restrictions*
- c) \(\boxed{k = \dfrac{13}{8}}\) — *No restrictions*
- g) \(\boxed{x = 4}\) — *Restriction: \(x \ne 2\)*

Let me know if you’d like the rest solved too!