\left|\frac{\left(3x-2\right)}{4}\right|-5=7

We need to solve the equation:

\[
\left|\frac{3x - 2}{4}\right| - 5 = 7
\]

### Step 1: Isolate the absolute value expression
First, add \(5\) to both sides to isolate the absolute value:

\[
\left|\frac{3x - 2}{4}\right| = 12
\]

### Step 2: Set up two equations
The absolute value equation \(\left|A\right| = B\) can be split into two cases:

1. \( \frac{3x - 2}{4} = 12 \)
2. \( \frac{3x - 2}{4} = -12 \)

### Step 3: Solve each equation

#### Case 1: \( \frac{3x - 2}{4} = 12 \)
Multiply both sides by \(4\):

\[
3x - 2 = 48
\]

Add \(2\) to both sides:

\[
3x = 50
\]

Divide by \(3\):

\[
x = \frac{50}{3}
\]

#### Case 2: \( \frac{3x - 2}{4} = -12 \)
Multiply both sides by \(4\):

\[
3x - 2 = -48
\]

Add \(2\) to both sides:

\[
3x = -46
\]

Divide by \(3\):

\[
x = \frac{-46}{3}
\]

### Step 4: Write the solution
The solutions to the equation are:

\[
x = \frac{50}{3} \quad \text{or} \quad x = \frac{-46}{3}
\]

Would you like further details or explanations?

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Here are 5 related questions:

1. How do you solve absolute value inequalities?
2. What is the graphical interpretation of absolute value equations?
3. Can you explain how to handle absolute values in a system of equations?
4. How does one check if a solution is valid in an absolute value equation?
5. What happens if the absolute value expression equals zero?

**Tip:** Always remember to check both positive and negative scenarios when dealing with absolute value equations!

Learn how to solve the absolute value equation |(3x-2)/4| - 5 = 7 with a step-by-step guide. This problem covers key algebraic concepts, including handling absolute values and solving linear equations. Suitable for students in Grades 9-10.

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