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The uploaded document contains multiple-choice inequality problems in Arabic. Let’s go through each problem one by one.

---

### Problem 1
Solve the inequality:
\[
3(x - 2) + 4 > 5(x + 3) - 5
\]

#### Solution:
1. Expand both sides:
   \[
   3x - 6 + 4 > 5x + 15 - 5
   \]
   \[
   3x - 2 > 5x + 10
   \]

2. Move terms involving \( x \) to one side and constants to the other:
   \[
   -2 > 2x + 10
   \]
   \[
   -12 > 2x
   \]

3. Divide by 2:
   \[
   x < -6
   \]

#### Answer: **Option (2)** \( x < -6 \)

---

### Problem 2
Identify the value of \( x \) that satisfies both inequalities:
\[
\frac{2x + 1}{3} > 5 \quad \text{and} \quad 2 - \frac{x}{2} < 3
\]

#### Solution:
1. Solve the first inequality:
   \[
   \frac{2x + 1}{3} > 5
   \]
   Multiply by 3:
   \[
   2x + 1 > 15
   \]
   \[
   2x > 14
   \]
   \[
   x > 7
   \]

2. Solve the second inequality:
   \[
   2 - \frac{x}{2} < 3
   \]
   Move 2 to the other side:
   \[
   -\frac{x}{2} < 1
   \]
   Multiply by -2 (reversing the inequality):
   \[
   x > -2
   \]

3. Combine both results:
   \[
   x > 7
   \]

#### Answer: **Option (2)** \( x > 7 \)

---

### Problem 3
Given \( x > 5 \), determine which of the following is necessarily true.

#### Solution:
Let’s analyze each option with the condition \( x > 5 \):
1. **Option (1):** \( 20 - 4x > x + 2 \)
2. **Option (2):** \( 4 + 2x > 54 \)
3. **Option (3):** \( 5x - 10 > 20 \)
4. **Option (4):** \( 3x + 4 < 19 \)

After testing each option, **Option (3)** \( 5x - 10 > 20 \) is correct when \( x > 5 \).

---

### Problem 4
Given \( c < 0 \), \( a < b \), and \( a, b, c > 0 \), identify the true statement.

#### Answer: **Option (3)** \( a \cdot b \cdot c < 0 \)

---

### Problem 5
Given \( a > 6 \), \( b < 12 \), identify the statement that **cannot** be true.

#### Answer: **Option (3)** \( b - a > 6 \)

Solve inequality problems involving algebraic expressions and comparison statements as seen in the uploaded document.

Explore solutions to multi-step algebraic inequalities, covering key concepts in inequality manipulation, solving systems of inequalities, and applying the properties of inequalities. Suitable for high school students in Grades 10-12.

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