https://file.aiquickdraw.com/mathbot/file/1726254651260h7oosywx.jpg

The image contains six mathematical equations involving absolute values and inequalities. Let's analyze each one:

1. **Q1:**  
   \[
   |x^2 + x + 1| + |2x - 3| = 4x
   \]
   This is an equation with two absolute value terms. To solve it, you'll need to break it into cases based on when the expressions inside the absolute values change signs.

2. **Q2:**  
   \[
   12x + 1 - 2|x - 2| = 5
   \]
   Again, you need to break this into cases based on when \( |x - 2| \) changes sign (i.e., when \( x = 2 \)).

3. **Q3:**  
   \[
   |x - 1| = |x - 2| + |x - 3|
   \]
   To solve this, you'll need to break it into several cases depending on the values of \(x\) relative to 1, 2, and 3, which affect the absolute value expressions.

4. **Q4:**  
   \[
   x^2 - x - |x - 3| - 10 \leq 0
   \]
   Similar to the previous questions, solve by considering cases based on when \(x = 3\), which affects the absolute value.

5. **Q5:**  
   \[
   \frac{x^2 + 6x - 7}{|x + 4|} < 0
   \]
   Here you will need to solve the inequality by considering both the quadratic numerator and the absolute value in the denominator.

6. **Q6:**  
   \[
   2|x + 1| = |x - 3| = 4
   \]
   This equation has two absolute value terms, requiring you to solve the system of equations by considering different cases.

Would you like me to solve any specific one or go through all of them step-by-step?

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### Five relative questions:
1. How do you solve absolute value equations by considering different cases?
2. What strategies help when solving absolute value inequalities?
3. What happens when both sides of an equation contain absolute values?
4. How do quadratic expressions impact inequalities involving absolute values?
5. Can you explain how to handle absolute values in rational expressions?

### Tip:
When solving absolute value equations, always determine where the expression inside the absolute value changes sign to split the solution into different cases.

Q1. |x^2 + x + 1| + |2x - 3| = 4x Q2. 12x + 1 - 2|x - 2| = 5 Q3. |x - 1| = |x - 2| + |x - 3| Q4. x^2 - x - |x - 3| - 10 ≤ 0 Q5. (x^2 + 6x - 7) / |x + 4| < 0 Q6. 2|x + 1| = |x - 3| = 4

Learn how to solve complex absolute value equations and inequalities, including quadratic expressions, with a step-by-step guide. This covers important mathematical concepts such as absolute value properties and quadratic equations, suitable for high school students in Grades 9-12.

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