Solve the following equations with absolute values in them.
`4|x+(1)/(2)|=18`

Answer

To solve the equation \( 4|x + \frac{1}{2}| = 18 \), we can follow these steps:

### Step 1: Isolate the Absolute Value
First, we need to isolate the absolute value on one side of the equation. We can do this by dividing both sides by 4:

\[ |x + \frac{1}{2}| = \frac{18}{4} \]

### Step 2: Simplify the Right Side
Next, we simplify \( \frac{18}{4} \):

\[ |x + \frac{1}{2}| = \frac{9}{2} \]

### Step 3: Set Up Two Cases
The absolute value equation \( |A| = B \) implies two cases: \( A = B \) and \( A = -B \). Therefore, we set up the two cases:

1. \( x + \frac{1}{2} = \frac{9}{2} \)
2. \( x + \frac{1}{2} = -\frac{9}{2} \)

### Step 4: Solve the First Case
For the first case:

\[ x + \frac{1}{2} = \frac{9}{2} \]

Subtract \( \frac{1}{2} \) from both sides:

\[ x = \frac{9}{2} - \frac{1}{2} = \frac{8}{2} = 4 \]

### Step 5: Solve the Second Case
For the second case:

\[ x + \frac{1}{2} = -\frac{9}{2} \]

Again, subtract \( \frac{1}{2} \) from both sides:

\[ x = -\frac{9}{2} - \frac{1}{2} = -\frac{10}{2} = -5 \]

### Step 6: Final Solutions
The solutions to the equation \( 4|x + \frac{1}{2}| = 18 \) are:

\[ x = 4 \quad \text{or} \quad x = -5 \]

### Summary of Steps
1. Isolate the absolute value by dividing by 4.
2. Simplify the right side.
3. Set up two cases based on the definition of absolute value.
4. Solve each case separately.
5. Combine the solutions.