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]
