https://file.aiquickdraw.com/mathbot/file/1726231174327bpqouvej.jpg

We are tasked with solving the absolute value inequality \( |3x - 4| > 12 \) and finding the solution in interval notation.

### Step 1: Break down the inequality
For the inequality \( |A| > B \), we can split it into two separate inequalities:
\[
A < -B \quad \text{or} \quad A > B
\]
In this case, \( A = 3x - 4 \) and \( B = 12 \). Thus, we have:
\[
3x - 4 < -12 \quad \text{or} \quad 3x - 4 > 12
\]

### Step 2: Solve both inequalities
#### Inequality 1: \( 3x - 4 < -12 \)
\[
3x < -12 + 4 \quad \Rightarrow \quad 3x < -8
\]
\[
x < \frac{-8}{3} \quad \Rightarrow \quad x < -\frac{8}{3}
\]

#### Inequality 2: \( 3x - 4 > 12 \)
\[
3x > 12 + 4 \quad \Rightarrow \quad 3x > 16
\]
\[
x > \frac{16}{3} \quad \Rightarrow \quad x > \frac{16}{3}
\]

### Step 3: Write the solution in interval notation
The solution to this inequality is:
\[
x < -\frac{8}{3} \quad \text{or} \quad x > \frac{16}{3}
\]
Thus, the solution in interval notation is:
\[
(-\infty, -\frac{8}{3}) \cup \left( \frac{16}{3}, \infty \right)
\]

### Step 4: Check the answer options
Now we compare this with the answer choices. We can see that the correct solution is represented as:
\[
\boxed{(-\infty, -\frac{8}{3}) \cup \left( \frac{16}{3}, \infty \right)}
\]
Therefore, the correct choice is the first option on the list.

Would you like further details or have any questions? Here are some related questions you can explore:

1. How do you solve absolute value inequalities of the form \( |A| \leq B \)?
2. What happens if the absolute value inequality is \( |3x - 4| \leq 12 \)?
3. How can absolute value inequalities be solved graphically?
4. How do we interpret the solution in a real-world context?
5. What happens when the inequality sign is reversed to \( |3x - 4| < 12 \)?

**Tip:** When solving absolute value inequalities, always split the inequality into two cases and remember to consider both positive and negative cases for the absolute value.

What is the solution (in interval notation) to the following absolute value inequality below? |3x - 4| > 12

Learn how to solve the absolute value inequality |3x - 4| > 12 in interval notation. This problem focuses on key algebraic concepts such as absolute value inequalities and their solutions in interval form. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation