To solve this inequality, we can examine the given expression step-by-step to confirm if the implication $|x + 2| < 0.3 \Rightarrow |4x + 8| < 1.2$ holds.

Step 1: Analyze the inequality $|x + 2| < 0.3$

This inequality implies: $-0.3 < x + 2 < 0.3$ Subtracting 2 from all parts: $-2.3 < x < -1.7$

Thus, the solution for $|x + 2| < 0.3$ is: $x \in (-2.3, -1.7)$

Step 2: Test the implication $|4x + 8| < 1.2$

We want to see if $x \in (-2.3, -1.7)$ guarantees that $|4x + 8| < 1.2$.

Rewrite $|4x + 8|$ using the fact that $4x + 8 = 4(x + 2)$: $|4x + 8| = |4(x + 2)| = 4 |x + 2|$

Since $|x + 2| < 0.3$, multiplying by 4 gives: $|4x + 8| < 4 \cdot 0.3 = 1.2$

Conclusion

Since $|4x + 8| < 1.2$ directly follows from $|x + 2| < 0.3$, the implication $|x + 2| < 0.3 \Rightarrow |4x + 8| < 1.2$ is true.

Would you like further clarification or have any questions?

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Here are some related questions you might find interesting:

1. What happens if we change the range in the initial inequality, say to $|x+2| < 0.5$?
2. How would the result change if we had $|x + 2| \leq 0.3$ instead?
3. Can this reasoning apply to any expression where a variable term is scaled?
4. What if $|x + 2| > 0.3$? Would the implication still hold?
5. How would the problem change if we had $|4x + 8| \leq 1.2$?

Tip: Inequalities often maintain their direction when multiplying by positive constants, but not when dealing with negatives—always verify signs when solving.