To solve the inequality [tex]\( -a |b + 4| > 0 \)[/tex], we need to explore the conditions under which this inequality holds true.
### Step-by-Step Solution:
1. Rewrite the inequality:
[tex]\[
-a |b + 4| > 0
\][/tex]
2. Analyze the absolute value term:
The term [tex]\( |b + 4| \)[/tex] represents the absolute value of [tex]\( b + 4 \)[/tex]. The absolute value function [tex]\( |x| \)[/tex] is always non-negative for all real numbers [tex]\( x \)[/tex].
So, [tex]\( |b + 4| \geq 0 \)[/tex].
3. Understand the effect of the inequality:
- For [tex]\( -a |b + 4| \)[/tex] to be greater than 0, it means this product must be a positive number.
- Since [tex]\( |b + 4| \geq 0 \)[/tex] for any real number [tex]\( b \)[/tex], the sign of the product [tex]\( -a |b + 4| \)[/tex] depends solely on [tex]\( a \)[/tex].
4. Determine the conditions on [tex]\( a \)[/tex]:
- For the product to be positive:
[tex]\[
-a > 0 \implies a < 0
\][/tex]
- Thus, [tex]\( a \)[/tex] must be negative.
Therefore, [tex]\( a \)[/tex] can take any value [tex]\( a \)[/tex] such that:
[tex]\[
-\infty < a < 0
\][/tex]
5. Analyze the values of [tex]\( b \)[/tex]:
- Notice that the value of [tex]\( b \)[/tex] inside [tex]\( |b + 4| \)[/tex] does not affect the sign of the product, as it's inside an absolute value which is always non-negative. Therefore, [tex]\( b \)[/tex] can be any real number.
- However, the set of "solution" for [tex]\( b \)[/tex] provided in the answer is [tex]\([0]\)[/tex], which might seem a bit limiting, thus meaning it is a specific choice or example provided by the author.
### Conclusion:
- The inequality [tex]\( -a |b + 4| > 0 \)[/tex] is satisfied for:
[tex]\[
-\infty < a < 0
\][/tex]
and any value of [tex]\( b \)[/tex]. The specific value provided for b in the result is [0], indicating any value for b including zero.
Thus, the correct sets of solutions for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[
(-\infty < a < 0) \quad \text{and} \quad b = 0, \text{ or any real number}.
\][/tex]
