Question 1171432
Let's solve these equations step-by-step.

**Equation 1: Log₂(3x - 7) + Log₂(x + 2) = Log₂(x + 1)**

1.  **Combine Logarithms:**
    * Using the property Log(a) + Log(b) = Log(ab), we get:
        * Log₂[(3x - 7)(x + 2)] = Log₂(x + 1)

2.  **Remove Logarithms:**
    * Since the logarithms have the same base, we can equate the arguments:
        * (3x - 7)(x + 2) = x + 1

3.  **Expand and Simplify:**
    * 3x² + 6x - 7x - 14 = x + 1
    * 3x² - x - 14 = x + 1
    * 3x² - 2x - 15 = 0

4.  **Solve the Quadratic Equation:**
    * We can factor the quadratic equation:
        * (3x + 5)(x - 3) = 0
    * This gives us two possible solutions:
        * 3x + 5 = 0  =>  x = -5/3
        * x - 3 = 0  =>  x = 3

5.  **Check for Valid Solutions:**
    * We need to ensure that the arguments of the logarithms are positive.
        * For x = -5/3:
            * 3x - 7 = 3(-5/3) - 7 = -12 (Negative, invalid)
            * x + 2 = -5/3 + 2 = 1/3 (Positive)
            * x + 1 = -5/3 + 1 = -2/3 (Negative, invalid)
        * For x = 3:
            * 3x - 7 = 3(3) - 7 = 2 (Positive)
            * x + 2 = 3 + 2 = 5 (Positive)
            * x + 1 = 3 + 1 = 4 (Positive)
    * Therefore, the only valid solution is x = 3.

**Solution for Equation 1: x = 3**

**Equation 2: Log₂(3x + 1) - Log₂(2 - 4x) > Log₂(5x - 2)**

1.  **Combine Logarithms:**
    * Using the property Log(a) - Log(b) = Log(a/b), we get:
        * Log₂[(3x + 1) / (2 - 4x)] > Log₂(5x - 2)

2.  **Remove Logarithms:**
    * Since the logarithms have the same base, we can remove the logarithms:
        * (3x + 1) / (2 - 4x) > 5x - 2

3.  **Solve the Inequality:**
    * First, we need to find the domain of the inequality:
        * 3x + 1 > 0  =>  x > -1/3
        * 2 - 4x > 0  =>  x < 1/2
        * 5x - 2 > 0  =>  x > 2/5
    * Combining these, the domain is 2/5 < x < 1/2.
    * Now, let's solve the inequality:
        * (3x + 1) > (5x - 2)(2 - 4x)
        * 3x + 1 > 10x - 20x² - 4 + 8x
        * 3x + 1 > 18x - 20x² - 4
        * 20x² - 15x + 5 > 0
        * 4x² - 3x + 1 > 0
    * The discriminant of the quadratic is:
        * (-3)² - 4(4)(1) = 9 - 16 = -7 (Negative)
    * Since the discriminant is negative, the quadratic is always positive.
    * Therefore, the inequality holds for all x in the domain 2/5 < x < 1/2.

**Solution for Equation 2: 2/5 < x < 1/2**