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) $b - a > 6$