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The image contains a series of inequalities to solve over \(\mathbb{R}\). Here's a transcription of the inequalities provided in the image:

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**Exercise 2: Solve the following inequalities in \(\mathbb{R}\):**

1. \((1 - \sqrt{2})x - 5 \leq 0\)  
2. \(3x - 5 < 7 - \sqrt{2}x\)  
3. \(\frac{7x - 2}{1 - \sqrt{3}} \leq \frac{7x + 2}{1 + \sqrt{3}}\)  
4. \(|x - 5| < \frac{1}{2}\)  
5. \(|x + 2| - |5| \leq 4\)  
6. \(|3x - 2| > 3\)

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Would you like me to solve these one by one and provide detailed explanations for each step? If yes, please let me know if you have a preference for which inequality to start with. 

Here are five related questions to consider:  
1. Would you like a general explanation of solving linear inequalities?  
2. Do you want assistance with absolute value inequality steps?  
3. Should we work on rational inequalities first?  
4. Do you need a graph or visualization of any of the solutions?  
5. Do you want me to check conditions of equality or boundaries?  

**Tip:** When solving inequalities, always consider simplifying terms and flipping the inequality sign if multiplying or dividing by a negative.

Solve the following inequalities in ℝ: 
1. (1 - √2)x - 5 ≤ 0 
2. 3x - 5 < 7 - √2x 
3. (7x - 2) / (1 - √3) ≤ (7x + 2) / (1 + √3) 
4. |x - 5| < 1/2 
5. |x + 2| - |5| ≤ 4 
6. |3x - 2| > 3

Learn how to solve advanced inequalities, including linear, rational, and absolute value inequalities. This detailed guide walks through each step with explanations, covering concepts such as cross-multiplication, boundary conditions, and properties of absolute values. Ideal for Grades 10-12.

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