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Let's solve each of the given equations step by step and then verify the solutions ("führe die Probe durch" means to perform the check).

### (a) Solve: 
\[
2 \cdot (x - 1) + x = 1 + 2 \cdot (x + 1)
\]

**Step 1:** Distribute the terms.
\[
2x - 2 + x = 1 + 2x + 2
\]

**Step 2:** Simplify both sides.
\[
3x - 2 = 2x + 3
\]

**Step 3:** Move all terms involving \(x\) to one side.
\[
3x - 2x = 3 + 2
\]
\[
x = 5
\]

**Step 4:** Verification (Probe):
Substitute \(x = 5\) into the original equation:
\[
2 \cdot (5 - 1) + 5 = 1 + 2 \cdot (5 + 1)
\]
\[
2 \cdot 4 + 5 = 1 + 2 \cdot 6
\]
\[
8 + 5 = 1 + 12
\]
\[
13 = 13
\]
The solution is correct.

### (b) Solve:
\[
7x - (2x + 3) = 3 \cdot (x + 1) + 2
\]

**Step 1:** Distribute the terms and simplify.
\[
7x - 2x - 3 = 3x + 3 + 2
\]
\[
5x - 3 = 3x + 5
\]

**Step 2:** Move all terms involving \(x\) to one side.
\[
5x - 3x = 5 + 3
\]
\[
2x = 8
\]

**Step 3:** Solve for \(x\).
\[
x = 4
\]

**Step 4:** Verification (Probe):
Substitute \(x = 4\) into the original equation:
\[
7 \cdot 4 - (2 \cdot 4 + 3) = 3 \cdot (4 + 1) + 2
\]
\[
28 - (8 + 3) = 3 \cdot 5 + 2
\]
\[
28 - 11 = 15 + 2
\]
\[
17 = 17
\]
The solution is correct.

### (c) Solve:
\[
\frac{x + 1}{2} - \frac{x - 1}{4} = x - 3
\]

**Step 1:** Find a common denominator on the left-hand side.
\[
\frac{2(x + 1)}{4} - \frac{x - 1}{4} = x - 3
\]
\[
\frac{2(x + 1) - (x - 1)}{4} = x - 3
\]

**Step 2:** Simplify the numerator.
\[
\frac{2x + 2 - x + 1}{4} = x - 3
\]
\[
\frac{x + 3}{4} = x - 3
\]

**Step 3:** Multiply both sides by 4 to eliminate the fraction.
\[
x + 3 = 4(x - 3)
\]

**Step 4:** Expand and simplify.
\[
x + 3 = 4x - 12
\]
\[
3 + 12 = 4x - x
\]
\[
15 = 3x
\]

**Step 5:** Solve for \(x\).
\[
x = 5
\]

**Step 6:** Verification (Probe):
Substitute \(x = 5\) into the original equation:
\[
\frac{5 + 1}{2} - \frac{5 - 1}{4} = 5 - 3
\]
\[
\frac{6}{2} - \frac{4}{4} = 2
\]
\[
3 - 1 = 2
\]
The solution is correct.

---

### Summary of Solutions:
- (a) \( x = 5 \)
- (b) \( x = 4 \)
- (c) \( x = 5 \)

Do you have any further questions or would you like more details on any step?

### Related Questions:
1. How can you solve equations involving fractions faster?
2. What are the steps for verifying solutions in algebra?
3. What are the common mistakes to avoid when solving linear equations?
4. How can we apply factoring methods to simplify more complex equations?
5. How can we solve systems of linear equations using substitution?

**Tip:** Always double-check your distribution and simplification steps, as small errors there can lead to incorrect solutions.

Solve the following equations and verify the solutions: a) 2 * (x - 1) + x = 1 + 2 * (x + 1), b) 7x - (2x + 3) = 3 * (x + 1) + 2, c) (x + 1) / 2 - (x - 1) / 4 = x - 3

Learn how to solve algebraic equations with distributive properties and fractions. This step-by-step guide solves three linear equations, explains the key concepts, and verifies the solutions. Suitable for students in Grades 8-10.

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