To solve the equation \( 6|1 - 5x| - 9 = 57 \), follow these steps:

### Step 1: Isolate the absolute value expression
First, add 9 to both sides of the equation:
\[
6|1 - 5x| = 66
\]

Next, divide both sides by 6:
\[
|1 - 5x| = 11
\]

### Step 2: Consider the two cases for the absolute value
The expression \( |1 - 5x| = 11 \) can result from two scenarios:

1. \( 1 - 5x = 11 \)
2. \( 1 - 5x = -11 \)

#### Case 1: \( 1 - 5x = 11 \)
Solve for \( x \):
\[
1 - 5x = 11
\]
Subtract 1 from both sides:
\[
-5x = 10
\]
Divide by -5:
\[
x = -2
\]

#### Case 2: \( 1 - 5x = -11 \)
Solve for \( x \):
\[
1 - 5x = -11
\]
Subtract 1 from both sides:
\[
-5x = -12
\]
Divide by -5:
\[
x = \frac{12}{5} = 2.4
\]

### Step 3: Write the solution
The solutions to the equation are:
\[
x = -2 \quad \text{or} \quad x = 2.4
\]

Would you like more details on any of these steps or have any questions?

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Here are some related questions you might find useful:

1. How do you solve an absolute value equation with more than one variable?
2. What happens if the absolute value expression includes fractions?
3. How can you check the solutions to an absolute value equation?
4. What are the properties of absolute value that are used in solving equations?
5. How would you solve an equation involving absolute value and exponents?
6. How can absolute value equations be represented graphically?
7. What is the general approach to solving inequalities with absolute values?
8. How does the solution of an absolute value equation change if the constant on the right side is negative?

**Tip:** Always check your solutions by substituting them back into the original equation to verify they satisfy the equation.

Learn how to solve the absolute value equation 6|1 - 5x| - 9 = 57 with a detailed step-by-step solution. This problem involves isolating the absolute value expression and solving for two cases. Suitable for high school students in Grades 9-12.

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