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Problem Breakdown:

Exercise 1:

1. Rewrite the sets in interval or union of intervals form:

   • $A = \{x \in \mathbb{R} \ | \ 5 - 4 \leq 2x + 4 \}$
   • $B = \{x \in \mathbb{R} \ | \ |2n - 4| \leq 2 \}$
   • $C = \{x \in \mathbb{R} \ | \ |3n - 2| \geq 5 \}$
   • $D = \{x \in \mathbb{R} \ | \ E(x + 4) = 3 \}$

2. Determine $A \cap B$, $A \cup B$, and $A \cap D$.

Exercise 2:

Solve the equations in $\mathbb{R}$:

1. $|1 - |1 - 4x|| = 5$
2. $|n - 4| = 2|4n - 2|$
3. $E(n - 4) = -3$.

Exercise 3:

Given $a$ and $b$ are real numbers such that $a \geq \frac{3}{2}$, $b \leq \frac{1}{2}$, and $a - b = 5$:

1. Prove $\frac{3}{2} \leq a \leq \frac{11}{2}$ and $-\frac{7}{2} \leq b \leq \frac{1}{2}$.
2. Calculate $A = \sqrt{|a - 3|} + \sqrt{|b - 4|}$.
3. Calculate $B = |a + b| + 2|a + b| + |a + b - 6|$.

Exercise 4:

Given $x$ is a real number such that $x \in ]-\frac{1}{4}, 1[$:

1. Verify $\frac{x^2}{1 + x} \leq 1 - x + \frac{x^2}{1 + x}$.
2. Prove $\frac{2}{3} \leq \frac{x}{2} + \frac{x^2}{1 + x} < 2$.

Exercise 5:

Given $a \in [3, a \cdot C]$:

1. Let $x = \sqrt{\frac{a - 1}{\sqrt{a + 1}}}$.
2. Verify certain inequalities (details provided in the sheet).
3. Deduce an approximate value for $\frac{3}{4}$, accurate to $10^{-4}$.

Exercise 6:

For $n \in \mathbb{N}^*$, let $A = \sqrt{\frac{a + x}{x}}$:

1. Prove $|A - \frac{1}{x}| \leq 3|x|$.
2. Deduce $A$, with precision $10^{-3}$.

Which of these exercises would you like me to solve or explain in detail first? Let me know if you'd like further clarification or breakdowns of specific sections!