Question

Solve: 2|x-4|+3>11
The answer has the form:
State your solution using interval notation like above:

Answer 1

To solve the inequality \( 2|x-4| + 3 > 11 \), we can start by isolating the absolute value expression.

1. Subtract 3 from both sides:

\[
2|x-4| > 8
\]

2. Divide both sides by 2:

\[
|x-4| > 4
\]

Next, we solve the inequality \( |x-4| > 4 \). This means that the expression inside the absolute value can either be greater than 4 or less than -4.

3. Therefore, we consider the two cases:

**Case 1**: \( x - 4 > 4 \)

\[
x > 8
\]

**Case 2**: \( x - 4 < -4 \)

\[
x < 0
\]

4. Now we combine the solutions from both cases. The solution to \( |x - 4| > 4 \) is:

\[
x < 0 \quad \text{or} \quad x > 8
\]

5. In interval notation, this solution can be represented as:

\[
(-\infty, 0) \cup (8, \infty)
\]

So, the final answer is:

\[
\boxed{(-\infty, 0) \cup (8, \infty)}
\]