Solve the folowing equations `|x|+2|x-6|=12`

Answer

To solve the equation \( |x| + 2|x-6| = 12 \), we will break it down into different cases based on the values of \( x \) that affect the absolute values.

### Step 1: Identify critical points
The absolute value expressions change at the points where their arguments are zero. For \( |x| \), this occurs at \( x = 0 \). For \( |x-6| \), this occurs at \( x = 6 \). Thus, we will consider three cases based on these critical points:
1. \( x < 0 \)
2. \( 0 \leq x < 6 \)
3. \( x \geq 6 \)

### Step 2: Case 1: \( x < 0 \)
In this case, both \( |x| \) and \( |x-6| \) will be negative:
\[ |x| = -x \quad \text{and} \quad |x-6| = -(x-6) = -x + 6 \]
Substituting these into the equation gives:
\[ -x + 2(-x + 6) = 12 \]
Simplifying this:
\[ -x - 2x + 12 = 12 \]
\[ -3x + 12 = 12 \]
Subtracting 12 from both sides:
\[ -3x = 0 \]
Dividing by -3:
\[ x = 0 \]
Since \( x = 0 \) does not satisfy \( x < 0 \), we discard this solution.

### Step 3: Case 2: \( 0 \leq x < 6 \)
In this case, \( |x| \) is positive and \( |x-6| \) is negative:
\[ |x| = x \quad \text{and} \quad |x-6| = -(x-6) = -x + 6 \]
Substituting these into the equation gives:
\[ x + 2(-x + 6) = 12 \]
Simplifying this:
\[ x - 2x + 12 = 12 \]
\[ -x + 12 = 12 \]
Subtracting 12 from both sides:
\[ -x = 0 \]
Dividing by -1:
\[ x = 0 \]
Since \( x = 0 \) is included in this range, we accept this solution.

### Step 4: Case 3: \( x \geq 6 \)
In this case, both \( |x| \) and \( |x-6| \) are positive:
\[ |x| = x \quad \text{and} \quad |x-6| = x - 6 \]
Substituting these into the equation gives:
\[ x + 2(x - 6) = 12 \]
Simplifying this:
\[ x + 2x - 12 = 12 \]
\[ 3x - 12 = 12 \]
Adding 12 to both sides:
\[ 3x = 24 \]
Dividing by 3:
\[ x = 8 \]
Since \( x = 8 \) satisfies \( x \geq 6 \), we accept this solution.

### Final Solutions
The solutions to the equation \( |x| + 2|x-6| = 12 \) are:
\[ x = 0 \quad \text{and} \quad x = 8 \]