The math problem in the image asks to solve the equation:

$14 - 9 | -7x + 1 | = 13$

To solve this equation, follow these steps:

Step 1: Isolate the absolute value term

First, move the constant term (14) to the other side of the equation:

$-9 | -7x + 1 | = 13 - 14$ $-9 | -7x + 1 | = -1$

Step 2: Simplify the equation

Divide both sides by -9 to isolate the absolute value expression:

$| -7x + 1 | = \frac{-1}{-9}$ $| -7x + 1 | = \frac{1}{9}$

Step 3: Set up two cases for the absolute value

The absolute value equation implies two possible scenarios:

1. $-7x + 1 = \frac{1}{9}$
2. $-7x + 1 = -\frac{1}{9}$

Step 4: Solve each case

Case 1: $-7x + 1 = \frac{1}{9}$

$-7x = \frac{1}{9} - 1 = \frac{1}{9} - \frac{9}{9} = -\frac{8}{9}$ $x = \frac{-\frac{8}{9}}{-7} = \frac{8}{63}$

Case 2: $-7x + 1 = -\frac{1}{9}$

$-7x = -\frac{1}{9} - 1 = -\frac{1}{9} - \frac{9}{9} = -\frac{10}{9}$ $x = \frac{-\frac{10}{9}}{-7} = \frac{10}{63}$

Final Solutions:

The solutions are:

$x = \frac{8}{63} \quad \text{or} \quad x = \frac{10}{63}$

Would you like further details or a clarification on any step? Here are some related questions to explore:

1. What is the significance of the absolute value in an equation?
2. How do you approach solving equations involving fractions?
3. Can you explain the concept of isolating terms in an equation?
4. What happens when the absolute value is set equal to a negative number?
5. How do you deal with multiple cases when solving absolute value equations?

Tip: Always remember to check if the solution satisfies the original equation by substituting it back in.