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 $A < -B$.