To solve the equation $\frac{|12 - x|}{2} = 4x$, follow these steps:

Step 1: Eliminate the denominator.

Multiply both sides by 2 to remove the fraction:

$|12 - x| = 8x$

Step 2: Consider the two cases for the absolute value.

Case 1: $12 - x \geq 0$

• Here, $|12 - x| = 12 - x$.
• The equation becomes:

$12 - x = 8x$

• Solve for $x$:

$12 = 9x \implies x = \frac{12}{9} = \frac{4}{3}$

• Check if this solution satisfies the condition $12 - x \geq 0$:

$12 - \frac{4}{3} = \frac{36}{3} - \frac{4}{3} = \frac{32}{3} > 0$

Since the condition is satisfied, $x = \frac{4}{3}$ is a valid solution.

Case 2: $12 - x < 0$

• Here, $|12 - x| = -(12 - x) = x - 12$.
• The equation becomes:

$x - 12 = 8x$

• Solve for $x$:

$-12 = 7x \implies x = -\frac{12}{7}$

• Check if this solution satisfies the condition $12 - x < 0$:

$12 - \left(-\frac{12}{7}\right) = 12 + \frac{12}{7} = \frac{84}{7} + \frac{12}{7} = \frac{96}{7} > 0$

This condition is not satisfied, so $x = -\frac{12}{7}$ is not a valid solution.

Conclusion:

The only valid solution is $x = \frac{4}{3}$.

Would you like further details or have any questions?

Here are some related questions:

1. How do you solve equations with absolute values?
2. What are the steps to check solutions in an absolute value equation?
3. How can you graph absolute value functions?
4. What happens if there are no valid solutions in an absolute value equation?
5. How does the distribution of inequalities affect the solution process?

Tip: When dealing with absolute value equations, always remember to consider both the positive and negative cases separately.