https://file.aiquickdraw.com/mathbot/file/1728756095563l93dft9e.jpg

The inequality in the image is:

\[
|21x - 2| < |7x + 1|
\]

To solve this absolute value inequality, we can break it into cases. The general rule is that for any two absolute values \( |A| < |B| \), we need to consider four scenarios:

1. \( A < B \)
2. \( A < -B \)
3. \( -A < B \)
4. \( -A < -B \)

Let's proceed step by step:

### Step 1: Break into cases
We'll consider the two main cases:

1. **Case 1: Both expressions are positive or zero.**
   \[
   21x - 2 < 7x + 1
   \]

2. **Case 2: One expression is positive, and the other is negative.**
   \[
   21x - 2 < -(7x + 1)
   \]

We'll solve each case separately and combine the results at the end.

#### Case 1: \( 21x - 2 < 7x + 1 \)

- Subtract \( 7x \) from both sides:
  \[
  21x - 7x - 2 < 1
  \]
  \[
  14x - 2 < 1
  \]

- Add 2 to both sides:
  \[
  14x < 3
  \]

- Divide by 14:
  \[
  x < \frac{3}{14}
  \]

#### Case 2: \( 21x - 2 < -(7x + 1) \)

- Distribute the negative sign:
  \[
  21x - 2 < -7x - 1
  \]

- Add \( 7x \) to both sides:
  \[
  21x + 7x - 2 < -1
  \]
  \[
  28x - 2 < -1
  \]

- Add 2 to both sides:
  \[
  28x < 1
  \]

- Divide by 28:
  \[
  x < \frac{1}{28}
  \]

### Combine the results:

From Case 1: \( x < \frac{3}{14} \)  
From Case 2: \( x < \frac{1}{28} \)

The stricter inequality is \( x < \frac{1}{28} \).

Thus, the solution is:

\[
x < \frac{1}{28}
\]

Let me know if you need any clarifications or further details.

---

Here are 5 related questions to expand your understanding:

1. How do we solve inequalities involving absolute values with different expressions?
2. Can we graph the solution of \( |21x - 2| < |7x + 1| \) on a number line?
3. How do we handle absolute value equations with equality instead of inequality?
4. What happens if one or both sides of the inequality are constants instead of expressions?
5. Can we check the validity of the solution by substituting values back into the inequality?

**Tip:** Always check for potential extraneous solutions when solving absolute value equations and inequalities by substituting the results back into the original inequality.

Learn how to solve the absolute value inequality |21x - 2| < |7x + 1| with a detailed step-by-step solution. This problem explores algebraic techniques and concepts, including handling absolute values and solving inequalities. Suitable for high school students in Grades 9-12.

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