Answer :

GinnyAnswer
To solve the absolute value inequality [tex]\(\frac{4|x+9|}{5} < 8\)[/tex], follow these steps:

1. Isolate the absolute value expression:
[tex]\[ \frac{4|x+9|}{5} < 8 \][/tex]
Multiply both sides by 5:
[tex]\[ 4|x+9| < 40 \][/tex]
Divide both sides by 4:
[tex]\[ |x+9| < 10 \][/tex]

2. Translate the absolute value inequality:
The inequality [tex]\(|x+9| < 10\)[/tex] means that the expression inside the absolute value lies within 10 units of zero. In other words, we can express this as:
[tex]\[ -10 < x+9 < 10 \][/tex]

3. Solve the compound inequality:
We need to solve both inequalities separately:

For the left part of the compound inequality:
[tex]\[ x+9 > -10 \][/tex]
Subtract 9 from both sides:
[tex]\[ x > -19 \][/tex]

For the right part of the compound inequality:
[tex]\[ x+9 < 10 \][/tex]
Subtract 9 from both sides:
[tex]\[ x < 1 \][/tex]

4. Combine the solutions:
The solution is the intersection of the two inequalities:
[tex]\[ -19 < x < 1 \][/tex]

Therefore, the inequalities are:

1. [tex]\(x < 1\)[/tex]
2. [tex]\(x > -19\)[/tex]

The solution is the interval:
[tex]\[ -19 < x < 1 \][/tex]

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