To solve the absolute value inequality [tex]\( |x + 5| < 4 \)[/tex], we can break it down into two separate inequalities according to the property of absolute values. Specifically, for any real number [tex]\( a \)[/tex], the inequality [tex]\( |a| < b \)[/tex] implies [tex]\( -b < a < b \)[/tex].
So, applying this to our problem:
[tex]\[ |x + 5| < 4 \][/tex]
This can be broken down into:
[tex]\[ -4 < x + 5 < 4 \][/tex]
Now, we solve the two inequalities separately.
1. Solve the inequality:
[tex]\[ -4 < x + 5 \][/tex]
Subtract 5 from both sides:
[tex]\[ -4 - 5 < x \][/tex]
[tex]\[ -9 < x \][/tex]
2. Solve the second inequality:
[tex]\[ x + 5 < 4 \][/tex]
Subtract 5 from both sides:
[tex]\[ x < 4 - 5 \][/tex]
[tex]\[ x < -1 \][/tex]
Combining these two results, we have:
[tex]\[ -9 < x < -1 \][/tex]
So, the solution to the inequality [tex]\( |x + 5| < 4 \)[/tex] is:
[tex]\[ -9 < x < -1 \][/tex]
Thus, the correct answer is:
a. [tex]\( -9 < x < -1 \)[/tex]
