https://file.aiquickdraw.com/mathbot/file/1727320864139o7vztxj2.jpg

The equation from the image is:

\[
4 | 3x - 6 | + 7 = 19
\]

Let's solve it step by step:

### Step 1: Isolate the absolute value expression.
\[
4 | 3x - 6 | + 7 = 19
\]
Subtract 7 from both sides:
\[
4 | 3x - 6 | = 12
\]
Divide both sides by 4:
\[
| 3x - 6 | = 3
\]

### Step 2: Solve the absolute value equation.
The equation \(| 3x - 6 | = 3\) can be split into two cases:

#### Case 1: \( 3x - 6 = 3 \)
Solve for \(x\):
\[
3x - 6 = 3
\]
Add 6 to both sides:
\[
3x = 9
\]
Divide by 3:
\[
x = 3
\]

#### Case 2: \( 3x - 6 = -3 \)
Solve for \(x\):
\[
3x - 6 = -3
\]
Add 6 to both sides:
\[
3x = 3
\]
Divide by 3:
\[
x = 1
\]

### Step 3: Final Solution
The solution set is:
\[
x = 1 \quad \text{or} \quad x = 3
\]

Would you like further details or have any questions?

### 5 related questions:
1. How do you handle equations with absolute values in general?
2. What happens if there is no solution to an absolute value equation?
3. How do you solve inequalities that involve absolute values?
4. What does it mean when an equation has more than one solution?
5. How can you check if your solution to an absolute value equation is correct?

### Tip:
Always check both cases when solving an absolute value equation, as it can lead to two potential solutions.

Solve the equation 4 | 3x - 6 | + 7 = 19.

Learn how to solve the absolute value equation 4 | 3x - 6 | + 7 = 19. This problem involves isolating the absolute value and solving two cases. Suitable for students in Grades 8-10.

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