Let's analyze the equations in the image and solve them step by step. These are absolute value equations and inequalities involving variable $x$. I will interpret each one and provide detailed solutions:

1. $|7 + x| = 2$ &\text{The absolute value equation implies:} \\ &7 + x = 2 \quad \text{or} \quad 7 + x = -2 \\ &\text{Solving each case:} \\ &x = 2 - 7 = -5 \quad \text{and} \quad x = -2 - 7 = -9 \\ &\boxed{x = -5 \text{ or } x = -9.} \end{aligned}$$
2. $|5 + x| = |2x + 1|$ &\text{This implies:} \\ &5 + x = 2x + 1 \quad \text{or} \quad 5 + x = -(2x + 1) \\ &\text{Case 1: } 5 + x = 2x + 1 \quad \Rightarrow \quad x = 4. \\ &\text{Case 2: } 5 + x = -2x - 1 \quad \Rightarrow \quad 3x = -6 \quad \Rightarrow \quad x = -2. \\ &\boxed{x = 4 \text{ or } x = -2.} \end{aligned}$$
3. $|3x - 4| < 3$ &\text{The inequality implies:} \\ &-3 < 3x - 4 < 3 \\ &\text{Solving: } -3 + 4 < 3x < 3 + 4 \quad \Rightarrow \quad 1 < 3x < 7 \\ &\text{Dividing through by 3: } \frac{1}{3} < x < \frac{7}{3}. \\ &\boxed{\frac{1}{3} < x < \frac{7}{3}.} \end{aligned}$$
4. $|5x - 1| \geq 4$ &\text{The inequality implies:} \\ &5x - 1 \geq 4 \quad \text{or} \quad 5x - 1 \leq -4 \\ &\text{Case 1: } 5x \geq 5 \quad \Rightarrow \quad x \geq 1. \\ &\text{Case 2: } 5x \leq -3 \quad \Rightarrow \quad x \leq -\frac{3}{5}. \\ &\boxed{x \geq 1 \text{ or } x \leq -\frac{3}{5}.} \end{aligned}$$
5. $|x^2 - 2x + 3| - 2 = 2$ &\text{Rewrite as: } |x^2 - 2x + 3| = 4. \\ &\text{This implies: } x^2 - 2x + 3 = 4 \quad \text{or} \quad x^2 - 2x + 3 = -4. \\ &\text{Case 1: } x^2 - 2x - 1 = 0. \\ &\text{Solve using quadratic formula: } x = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2}. \\ &\text{Case 2: } x^2 - 2x + 7 = 0 \quad \text{(No real solutions, as discriminant is negative).} \\ &\boxed{x = 1 + \sqrt{2} \text{ or } x = 1 - \sqrt{2}.} \end{aligned}$$
6. $|\frac{2x}{7} + 3| \geq \frac{5}{7}$ &\text{The inequality implies:} \\ &\frac{2x}{7} + 3 \geq \frac{5}{7} \quad \text{or} \quad \frac{2x}{7} + 3 \leq -\frac{5}{7} \\ &\text{Case 1: } \frac{2x}{7} \geq -\frac{16}{7} \quad \Rightarrow \quad x \geq -8. \\ &\text{Case 2: } \frac{2x}{7} \leq -\frac{26}{7} \quad \Rightarrow \quad x \leq -13. \\ &\boxed{x \geq -8 \text{ or } x \leq -13.} \end{aligned}$$

Would you like further clarification on any solution? Here are five related questions:

1. How do we graph absolute value equations or inequalities on a number line?
2. What are the common methods to solve quadratic equations like in question 5?
3. How can inequalities be solved when involving fractions or multiple variables?
4. What are the properties of absolute values used in these solutions?
5. How can we verify the solutions of absolute value equations?

Tip: Always check your solutions by substituting them back into the original equation or inequality to ensure correctness.