1. To solve the equation (2x - 7)^3 = 27/63, we can start by taking the cube root of both sides:

∛((2x - 7)^3) = ∛(27/63)

2x - 7 = ∛(27/63)

Next, simplify the cube root:

2x - 7 = ∛(3/7)

Now, cube both sides to eliminate the cube root:

(2x - 7)^3 = (∛(3/7))^3

(2x - 7)^3 = 3/7

Expand the left side:

8x^3 - 84x^2 + 294x - 343 = 3/7

Multiply both sides by 7 to get rid of the fraction:

56x^3 - 588x^2 + 2058x - 2401 = 3

Rearrange the equation:

56x^3 - 588x^2 + 2058x - 2404 = 0

This is a cubic equation that can be solved using various methods such as factoring, synthetic division, or using a graphing calculator.

2. To solve the equation (7 - 3x)^2 = 25/144, we can start by taking the square root of both sides:

√((7 - 3x)^2) = √(25/144)

7 - 3x = √(25/144)

Next, simplify the square root:

7 - 3x = √(25)/√(144)

7 - 3x = 5/12

Now, isolate the variable:

-3x = 5/12 - 7

-3x = 5/12 - 84/12

-3x = -79/12

Finally, divide both sides by -3 to solve for x:

x = (-79/12) / -3

x = 79/36

3. To solve the equation |2x - 3| = x + 1, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 2x - 3 is positive (2x - 3 > 0)

In this case, the equation becomes:

2x - 3 = x + 1

Simplify and solve for x:

2x - x = 1 + 3

x = 4

Case 2: 2x - 3 is negative (2x - 3 < 0)

In this case, the equation becomes:

-(2x - 3) = x + 1

Simplify and solve for x:

-2x + 3 = x + 1

-3 - 1 = x + 2x

-4 = 3x

x = -4/3

Therefore, the solutions to the equation |2x - 3| = x + 1 are x = 4 and x = -4/3.

4. To solve the equation |3x + 1| = 5, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 3x + 1 is positive (3x + 1 > 0)

In this case, the equation becomes:

3x + 1 = 5

Simplify and solve for x:

3x = 5 - 1

3x = 4

x = 4/3

Case 2: 3x + 1 is negative (3x + 1 < 0)

In this case, the equation becomes:

-(3x + 1) = 5

Simplify and solve for x:

-3x - 1 = 5

-3x = 5 + 1

-3x = 6

x = -6/3

x = -2

Therefore, the solutions to the equation |3x + 1| = 5 are x = 4/3 and x = -2.