3. (2(x -3) ≤ 3x + 4(x +1), ((x -3) (x +3) ≤ (x -4)ª -1; 4. (2(x + 11) ≥ 3(6 - x), (x -3) (x +6) ≥ (x +5) (x -4);

Let's solve the inequalities step-by-step.

Problem 3:

Inequality 1: \(2(x - 3) \leq 3x + 4(x + 1)\)

1. Distribute: \[ 2x - 6 \leq 3x + 4x + 4 \] \[ 2x - 6 \leq 7x + 4 \]

2. Rearranging terms: \[ 2x - 7x \leq 4 + 6 \] \[ -5x \leq 10 \]

3. Dividing by -5 (remember to flip the inequality sign): \[ x \geq -2 \]

Inequality 2: \((x - 3)(x + 3) \leq (x - 4)^2 - 1\)

1. Expand both sides: \[ x^2 - 9 \leq (x^2 - 8x + 16) - 1 \] \[ x^2 - 9 \leq x^2 - 8x + 15 \]

2. Subtract \(x^2\) from both sides: \[ -9 \leq -8x + 15 \]

3. Rearranging terms: \[ -8x \geq -24 \]

4. Dividing by -8 (flip the inequality): \[ x \leq 3 \]

Combined solution for Problem 3:

The solution set for the first inequality is \(x \geq -2\) and for the second inequality is \(x \leq 3\). Therefore, the combined solution is: \[ -2 \leq x \leq 3 \]

Problem 4:

Inequality 1: \(2(x + 11) \geq 3(6 - x)\)

1. Distributing: \[ 2x + 22 \geq 18 - 3x \]

2. Rearranging terms: \[ 2x + 3x \geq 18 - 22 \] \[ 5x \geq -4 \]

3. Dividing by 5: \[ x \geq -\frac{4}{5} \]

Inequality 2: \((x - 3)(x + 6) \geq (x + 5)(x - 4)\)

1. Expand both sides: \[ x^2 + 3x - 18 \geq x^2 + x - 20 \]

2. Subtract \(x^2\) from both sides: \[ 3x - 18 \geq x - 20 \]

3. Rearranging terms: \[ 3x - x \geq -20 + 18 \] \[ 2x \geq -2 \]

4. Dividing by 2: \[ x \geq -1 \]

Combined solution for Problem 4:

The solution set for the first inequality is \(x \geq -\frac{4}{5}\) and for the second inequality is \(x \geq -1\). Therefore, the combined solution is: \[ x \geq -\frac{4}{5} \]

Summary of both problems:

• Problem 3: \( -2 \leq x \leq 3 \)
• Problem 4: \( x \geq -\frac{4}{5} \)