https://file.aiquickdraw.com/mathbot/file/1726065424478bhv8jbox.jpg

The image contains a series of inequalities involving polynomial and factored expressions. I'll help you solve each of them step by step. Let's break down the inequalities:

1. **Problem a**: \((x - 1)^2 \cdot (x - 2)^3 \cdot (x - 3)^4 \cdot (x - 4)^5 \geq 0\)

2. **Problem b**: \((x - 1)^5 \cdot (x + 2)^3 \cdot (x + 4)^2 < 0\)

3. **Problem c**: \((x - 2)(x - 3)(x - 5)(x - 6) < 0\)

4. **Problem d**: \((x^2 - 36)(x^2 - 8x + 7) < 0\)

5. **Problem e**: \(x^4 - 3x^2 - x + 3 < 0\)

To solve these inequalities, we'll use a method called the **sign chart** (interval method). Here's a general approach for each:

### 1. Problem a: \((x - 1)^2 \cdot (x - 2)^3 \cdot (x - 3)^4 \cdot (x - 4)^5 \geq 0\)

- Identify roots: \(x = 1, 2, 3, 4\)
- Analyze the sign of each factor in different intervals by considering the exponents.
- Roots with even powers do not change the sign of the expression at that root, while roots with odd powers do.
  
### 2. Problem b: \((x - 1)^5 \cdot (x + 2)^3 \cdot (x + 4)^2 < 0\)

- Identify roots: \(x = 1, -2, -4\)
- Check sign changes across intervals.

### 3. Problem c: \((x - 2)(x - 3)(x - 5)(x - 6) < 0\)

- Roots: \(x = 2, 3, 5, 6\)
- Analyze the behavior across the intervals formed by the roots.

### 4. Problem d: \((x^2 - 36)(x^2 - 8x + 7) < 0\)

- Factor further: \(x^2 - 36 = (x - 6)(x + 6)\) and \(x^2 - 8x + 7 = (x - 7)(x - 1)\)
- Roots: \(x = -6, -1, 1, 7\)

### 5. Problem e: \(x^4 - 3x^2 - x + 3 < 0\)

- For higher-degree polynomials, numerical or graphical methods are often more effective to analyze the roots and intervals.

Would you like to go into detail for each problem, or is there a specific one you'd like to start with? Let me know if you'd like step-by-step solutions for any of these!

Solve the inequalities in the provided image involving polynomial and factored expressions.

Explore how to solve polynomial inequalities like (x - 1)^2 · (x - 2)^3 · (x - 3)^4 · (x - 4)^5 ≥ 0 using sign chart analysis, factoring techniques, and interval testing. Suitable for high school students learning advanced algebra.

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