To solve the inequality:

\[
2\left|x+6\right| + 4 \le 20
\]

we follow these steps:

### Step 1: Isolate the Absolute Value

First, subtract 4 from both sides:

\[
2\left|x+6\right| \le 16
\]

Then, divide both sides by 2:

\[
\left|x+6\right| \le 8
\]

### Step 2: Remove the Absolute Value

The absolute value inequality \(\left|x+6\right| \le 8\) means that the expression inside the absolute value, \(x+6\), must be between \(-8\) and \(8\). This gives us two inequalities:

1. \(x + 6 \leq 8\)
2. \(x + 6 \geq -8\)

### Step 3: Solve Each Inequality

1. **For \(x + 6 \leq 8\):**

   Subtract 6 from both sides:

   \[
   x \leq 2
   \]

2. **For \(x + 6 \geq -8\):**

   Subtract 6 from both sides:

   \[
   x \geq -14
   \]

### Step 4: Write the Solution as an Interval

Combining both inequalities, we get:

\[
-14 \leq x \leq 2
\]

### Final Answer

The solution in interval notation is:

\[
\boxed{[-14, 2]}
\]

Would you like more details, or do you have any questions?

### Related Questions

1. How do you solve inequalities involving absolute values with more complex expressions?
2. How would the solution change if the inequality were strict, such as \(2\left|x+6\right| + 4 < 20\)?
3. Can absolute value inequalities have no solution? If so, under what conditions?
4. How do you graph the solution \([-14, 2]\) on a number line?
5. How do you solve absolute value inequalities where the expression inside is quadratic?

### Tip

When dealing with absolute value inequalities, always remember that \(|A| \leq B\) can be rewritten as \(-B \leq A \leq B\), and \(|A| > B\) can be split into two separate cases: \(A > B\) or \(A < -B\).

Learn how to solve the absolute value inequality 2|x + 6| + 4 ≤ 20 with a detailed step-by-step solution. This problem covers key algebraic concepts, including absolute value properties and inequalities. Suitable for students in Grades 9-10.

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