Thanks for sharing the image! I'll analyze the exercises and solve them one by one. Let’s get started:

Exercise 1

Rewrite the expressions without using the absolute value symbol:

• (a) $A = |6x - 19|$
  Rewrite using a piecewise definition:

  \begin{cases} 6x - 19, & \text{if } 6x - 19 \geq 0 \\ -(6x - 19), & \text{if } 6x - 19 < 0 \end{cases}$$

• (b) $B = |5\sqrt{5} - 8\sqrt{2}|$
  Calculate the value of $5\sqrt{5}$ and $8\sqrt{2}$ to compare: $5\sqrt{5} \approx 11.18, \quad 8\sqrt{2} \approx 11.31$ Since $8\sqrt{2} > 5\sqrt{5}$: $B = -(5\sqrt{5} - 8\sqrt{2}) = 8\sqrt{2} - 5\sqrt{5}$

• (c) $C = |2x - 4|$
  Similar to (a):

  \begin{cases} 2x - 4, & \text{if } 2x - 4 \geq 0 \\ -(2x - 4), & \text{if } 2x - 4 < 0 \end{cases}$$

Exercise 2

(1) Solve graphically:
For $|x - 3| = 2$ and $|x + 2| = 3$, the solutions correspond to the points where the distance from $x$ to the given center equals the constant.

• $|x - 3| = 2$:
  $x - 3 = 2 \quad \text{or} \quad x - 3 = -2 \implies x = 5 \quad \text{or} \quad x = 1$

• $|x + 2| = 3$:
  $x + 2 = 3 \quad \text{or} \quad x + 2 = -3 \implies x = 1 \quad \text{or} \quad x = -5$

(2) Solve algebraically:
These steps are already included above.

Exercise 3

(1) Solve equations:
Solve each equation step by step.

• $\frac{2x - 3}{7} = 2$:
  Multiply through by 7: $2x - 3 = 14 \implies 2x = 17 \implies x = \frac{17}{2}$

• $|3x + 5| = |-2x + 3|$:
  Consider cases:
  (a) $3x + 5 = -2x + 3$, (b) $3x + 5 = 2x - 3$.

• $|x - 6| = -9$:
  No solution (absolute values are non-negative).

• $|x + 8| = 0$:
  Solve $x + 8 = 0$: $x = -8$

(2) Solve inequalities:
Proceed as in Exercise 1 by splitting into cases.

Exercise 4

Determine $I \cap J$ and $I \cup J$ for the intervals.

Exercise 5

(1) Simplify expressions:
Evaluate the given square roots.

(2) Solve inequalities step-by-step.

Would you like me to work through any specific exercise in more detail? Let me know!

Questions to deepen understanding:

1. What are the properties of absolute value that help in solving equations?
2. How do piecewise functions simplify absolute value problems?
3. Can you graph the equations in Exercise 2 on a number line?
4. Why is $|x - 6| = -9$ unsolvable in real numbers?
5. How does interval arithmetic work in union and intersection problems?

Tip:

For absolute value equations, always split into cases based on the definition of $|x|$.