Solve `|2x+6|le16`.

Answer

To solve the inequality \( |2x + 6| \leq 16 \), we will follow these steps:

### Step 1: Understand the absolute value inequality
The absolute value inequality \( |y| \leq a \) means that \( -a \leq y \leq a \). In our case, \( y = 2x + 6 \) and \( a = 16 \).

### Step 2: Set up the two inequalities
From the absolute value inequality, we can write:
\[ -16 \leq 2x + 6 \leq 16 \]

### Step 3: Break it into two separate inequalities
We will solve the two inequalities separately:
1. \( 2x + 6 \leq 16 \)
2. \( 2x + 6 \geq -16 \)

### Step 4: Solve the first inequality
For the first inequality \( 2x + 6 \leq 16 \):
\[ 2x \leq 16 - 6 \]
\[ 2x \leq 10 \]
\[ x \leq \frac{10}{2} \]
\[ x \leq 5 \]

### Step 5: Solve the second inequality
For the second inequality \( 2x + 6 \geq -16 \):
\[ 2x \geq -16 - 6 \]
\[ 2x \geq -22 \]
\[ x \geq \frac{-22}{2} \]
\[ x \geq -11 \]

### Step 6: Combine the results
From the two inequalities, we have:
\[ -11 \leq x \leq 5 \]

### Step 7: Write the final solution
The solution in interval notation is:
\[ [-11, 5] \]