https://r2.erweima.ai/mathbot/image/1726340263934lz4rzfp4.jpg

The image shows a compound inequality which reads:

\[ |x + 2| < 0.3 \implies |4x + 8| < 1.2 \]

To solve this, we start by addressing the first inequality and then check the implication for the second inequality.

### Step 1: Solve \( |x + 2| < 0.3 \)

This absolute value inequality means that \(x + 2\) must be within 0.3 units of 0, hence:
\[ -0.3 < x + 2 < 0.3 \]
Subtracting 2 from each part:
\[ -2.3 < x < -1.7 \]

### Step 2: Verify the implication \( |4x + 8| < 1.2 \)

To ensure the implication holds true, we check the result from Step 1. Substituting \(x\) from the range \(-2.3\) to \(-1.7\):
1. For \(x = -2.3\):
   \[ 4(-2.3) + 8 = -1.2 \]
   Hence, \( |4(-2.3) + 8| = 1.2 \) which is exactly on the boundary.

2. For \(x = -1.7\):
   \[ 4(-1.7) + 8 = -0.8 \]
   Hence, \( |4(-1.7) + 8| = 0.8 \)

Since all values within \(-2.3 < x < -1.7\) result in \( |4x + 8| \leq 1.2\), the implication holds.

Thus, the solution for \(x\) that satisfies both conditions of the compound inequality is:
\[ -2.3 < x < -1.7 \]

Would you like more details or have any other questions about this problem?

Here are some additional queries to consider:
1. How would the solution change if the inequalities were non-strict (i.e., \(\leq\) instead of \(<\))?
2. What is the general approach to solve any compound inequality involving absolute values?
3. How do the results vary with different coefficients inside the absolute value brackets?
4. Can you solve for \(x\) if only the second inequality is given?
5. What are the implications of shifting the values inside the absolute brackets by a constant?

**Tip:** When dealing with absolute value inequalities, always consider breaking it into two separate cases based on the definition of absolute value.

Learn how to solve a compound absolute value inequality involving two inequalities. We break down the steps and provide detailed explanations suitable for Grades 9-12.

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