To solve the absolute value inequality:
[tex]\[
\frac{4|x+9|}{5} < 8
\][/tex]
we can follow these steps:
1. Isolate the absolute value:
[tex]\[
\frac{4|x+9|}{5} < 8
\][/tex]
First, clear the fraction by multiplying both sides by [tex]\(5\)[/tex]:
[tex]\[
4|x+9| < 40
\][/tex]
Next, divide both sides by [tex]\(4\)[/tex]:
[tex]\[
|x+9| < 10
\][/tex]
2. Split the absolute value into two inequalities:
The expression [tex]\(|x+9| < 10\)[/tex] indicates that the value inside the absolute value, [tex]\(x+9\)[/tex], must be between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex]. Therefore, we can write:
[tex]\[
-10 < x + 9 < 10
\][/tex]
3. Solve the compound inequality:
To isolate [tex]\(x\)[/tex], subtract [tex]\(9\)[/tex] from all parts of the inequality:
[tex]\[
-10 - 9 < x + 9 - 9 < 10 - 9
\][/tex]
Simplifying the inequality, we get:
[tex]\[
-19 < x < 1
\][/tex]
Therefore, the solution to the inequality [tex]\(\frac{4|x+9|}{5} < 8\)[/tex] is:
[tex]\[
-19 < x < 1
\][/tex]
In other words,
[tex]\[
x > -19 \quad \text{and} \quad x < 1
\][/tex]
