https://file.aiquickdraw.com/mathbot/file/172680414838569raqebf.jpg

The equation in the image is:

\[
|3x - 4| - 4 = 2x
\]

Let's solve this equation for all values of \(x\).

### Step 1: Isolate the absolute value
\[
|3x - 4| = 2x + 4
\]

### Step 2: Handle the absolute value
Since the absolute value has two cases, we need to solve for both possibilities:

#### Case 1: \(3x - 4 = 2x + 4\)
\[
3x - 2x = 4 + 4
\]
\[
x = 8
\]

#### Case 2: \(3x - 4 = -(2x + 4)\)
\[
3x - 4 = -2x - 4
\]
\[
3x + 2x = -4 + 4
\]
\[
5x = 0
\]
\[
x = 0
\]

### Step 3: Verify solutions
We need to check if both solutions satisfy the original equation.

- For \(x = 8\):
  \[
  |3(8) - 4| - 4 = 2(8)
  \]
  \[
  |24 - 4| - 4 = 16
  \]
  \[
  |20| - 4 = 16
  \]
  \[
  20 - 4 = 16
  \quad \text{(True)}
  \]

- For \(x = 0\):
  \[
  |3(0) - 4| - 4 = 2(0)
  \]
  \[
  |-4| - 4 = 0
  \]
  \[
  4 - 4 = 0
  \quad \text{(True)}
  \]

Both solutions \(x = 8\) and \(x = 0\) are valid.

### Final Answer:
The solutions are \(x = 8\) and \(x = 0\).

Would you like any further clarification or additional details?

Here are five related questions to explore further:
1. How do we approach solving an equation with absolute values in general?
2. Can there be cases where no solution exists for an absolute value equation?
3. What is the geometric interpretation of absolute value in equations?
4. How does solving an absolute value inequality differ from solving an equation?
5. Can you solve a similar equation but with different coefficients?

**Tip:** Always check your solutions in absolute value equations to ensure they satisfy the original equation.

Solve the equation for all values of x: |3x - 4| - 4 = 2x

Learn how to solve the absolute value equation |3x - 4| - 4 = 2x with step-by-step instructions. This solution covers key algebraic concepts, including handling absolute values and solving linear equations. Suitable for students in Grades 9-11.

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